diff --git a/_forest b/_forest
index be8ffe98..0184973a 160000
--- a/_forest
+++ b/_forest
@@ -1 +1 @@
-Subproject commit be8ffe98cb7733cfae42ee64fc0335d14499b43e
+Subproject commit 0184973a56546201b8891e5e35dffe29a9f1eccc
diff --git a/_site/2023/04/09/gsi-applications.html b/_site/2023/04/09/gsi-applications.html
index 162f1786..b0270cea 100644
--- a/_site/2023/04/09/gsi-applications.html
+++ b/_site/2023/04/09/gsi-applications.html
@@ -83,8 +83,8 @@ <h2 id="engr-101-f23">ENGR 101 (F23)</h2>
 
 <iframe width="560" height="315" src="https://www.youtube.com/embed/Cd9Tq7s_BhQ" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen=""></iframe>
 
-<style> .pdf-embed-wrap-640a4ed8-8b93-4985-9785-bfb133703a8d { display: flex; flex-direction: column; width: 100%; height: 650px; } .pdf-embed-container-640a4ed8-8b93-4985-9785-bfb133703a8d { height: 100%; } .pdf-link-640a4ed8-8b93-4985-9785-bfb133703a8d { background-color: white; text-align: center; border-style: solid; } .pdf-embed-container-640a4ed8-8b93-4985-9785-bfb133703a8d iframe { width: 100%; height: 100%; } </style>
-<div class="pdf-embed-wrap-640a4ed8-8b93-4985-9785-bfb133703a8d"> <div class="pdf-link-640a4ed8-8b93-4985-9785-bfb133703a8d"> <a href="/files/101-gsi-application.pdf" target="_blank">View PDF</a> </div> <div class="pdf-embed-container-640a4ed8-8b93-4985-9785-bfb133703a8d"> <iframe src="/files/101-gsi-application.pdf" frameborder="0" allowfullscreen=""></iframe> </div> </div>
+<style> .pdf-embed-wrap-798aab41-27ff-4376-bab4-3e0f273277b2 { display: flex; flex-direction: column; width: 100%; height: 650px; } .pdf-embed-container-798aab41-27ff-4376-bab4-3e0f273277b2 { height: 100%; } .pdf-link-798aab41-27ff-4376-bab4-3e0f273277b2 { background-color: white; text-align: center; border-style: solid; } .pdf-embed-container-798aab41-27ff-4376-bab4-3e0f273277b2 iframe { width: 100%; height: 100%; } </style>
+<div class="pdf-embed-wrap-798aab41-27ff-4376-bab4-3e0f273277b2"> <div class="pdf-link-798aab41-27ff-4376-bab4-3e0f273277b2"> <a href="/files/101-gsi-application.pdf" target="_blank">View PDF</a> </div> <div class="pdf-embed-container-798aab41-27ff-4376-bab4-3e0f273277b2"> <iframe src="/files/101-gsi-application.pdf" frameborder="0" allowfullscreen=""></iframe> </div> </div>
 <h2 id="eecs-376-f23">EECS 376 (F23)</h2>
 <p><strong>outcome</strong>: rejected. No response.</p>
 <h3 id="too-long-version">too long version</h3>
@@ -153,7 +153,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/04/10/miserliness.html b/_site/2023/04/10/miserliness.html
index 8e61c2de..cb10548d 100644
--- a/_site/2023/04/10/miserliness.html
+++ b/_site/2023/04/10/miserliness.html
@@ -142,7 +142,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/04/14/rasterize-pdfs.html b/_site/2023/04/14/rasterize-pdfs.html
index 88d9d78c..433bde2f 100644
--- a/_site/2023/04/14/rasterize-pdfs.html
+++ b/_site/2023/04/14/rasterize-pdfs.html
@@ -135,7 +135,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/04/23/scripting.html b/_site/2023/04/23/scripting.html
index 282bb054..13eaac0b 100644
--- a/_site/2023/04/23/scripting.html
+++ b/_site/2023/04/23/scripting.html
@@ -266,7 +266,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/05/02/chili-oil.html b/_site/2023/05/02/chili-oil.html
index bb135b4a..87370b16 100644
--- a/_site/2023/05/02/chili-oil.html
+++ b/_site/2023/05/02/chili-oil.html
@@ -122,7 +122,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/05/07/nonstandard-analysis.html b/_site/2023/05/07/nonstandard-analysis.html
index 688a9388..e8790c60 100644
--- a/_site/2023/05/07/nonstandard-analysis.html
+++ b/_site/2023/05/07/nonstandard-analysis.html
@@ -116,7 +116,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/05/10/linux-graphics-modules.html b/_site/2023/05/10/linux-graphics-modules.html
index 850cc3a9..47de6cfa 100644
--- a/_site/2023/05/10/linux-graphics-modules.html
+++ b/_site/2023/05/10/linux-graphics-modules.html
@@ -137,7 +137,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/07/24/book-catalogue.html b/_site/2023/07/24/book-catalogue.html
index cdbe73b0..594b4f58 100644
--- a/_site/2023/07/24/book-catalogue.html
+++ b/_site/2023/07/24/book-catalogue.html
@@ -291,7 +291,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/08/03/nic-bonding.html b/_site/2023/08/03/nic-bonding.html
index 2332f138..e5db1cbb 100644
--- a/_site/2023/08/03/nic-bonding.html
+++ b/_site/2023/08/03/nic-bonding.html
@@ -126,7 +126,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/10/10/tortoise-tts.html b/_site/2023/10/10/tortoise-tts.html
index 5e9ceebd..f03f688a 100644
--- a/_site/2023/10/10/tortoise-tts.html
+++ b/_site/2023/10/10/tortoise-tts.html
@@ -252,7 +252,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/10/29/coordinating-processes.html b/_site/2023/10/29/coordinating-processes.html
index 6061032f..e15e2860 100644
--- a/_site/2023/10/29/coordinating-processes.html
+++ b/_site/2023/10/29/coordinating-processes.html
@@ -108,7 +108,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/11/07/ble-sh.html b/_site/2023/11/07/ble-sh.html
index 91467875..a83fe09e 100644
--- a/_site/2023/11/07/ble-sh.html
+++ b/_site/2023/11/07/ble-sh.html
@@ -106,7 +106,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/12/19/icecc.html b/_site/2023/12/19/icecc.html
index d4584883..10696c9f 100644
--- a/_site/2023/12/19/icecc.html
+++ b/_site/2023/12/19/icecc.html
@@ -131,7 +131,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/12/20/xmonad.html b/_site/2023/12/20/xmonad.html
index eabee125..82cd2731 100644
--- a/_site/2023/12/20/xmonad.html
+++ b/_site/2023/12/20/xmonad.html
@@ -193,7 +193,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/01/08/thunderbird-and-nix.html b/_site/2024/01/08/thunderbird-and-nix.html
index b6d0a0a7..66c6c644 100644
--- a/_site/2024/01/08/thunderbird-and-nix.html
+++ b/_site/2024/01/08/thunderbird-and-nix.html
@@ -170,7 +170,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/01/16/mellanox.html b/_site/2024/01/16/mellanox.html
index 34adc895..96504baf 100644
--- a/_site/2024/01/16/mellanox.html
+++ b/_site/2024/01/16/mellanox.html
@@ -110,7 +110,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/02/08/natural-models.html b/_site/2024/02/08/natural-models.html
index 2ad6cafb..2c7f3037 100644
--- a/_site/2024/02/08/natural-models.html
+++ b/_site/2024/02/08/natural-models.html
@@ -166,7 +166,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/02/12/noctua.html b/_site/2024/02/12/noctua.html
index a0c76431..ac63e03d 100644
--- a/_site/2024/02/12/noctua.html
+++ b/_site/2024/02/12/noctua.html
@@ -115,7 +115,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/02/24/vim.html b/_site/2024/02/24/vim.html
index 7d4fead7..5053ff9e 100644
--- a/_site/2024/02/24/vim.html
+++ b/_site/2024/02/24/vim.html
@@ -123,7 +123,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/03/10/deoplete.html b/_site/2024/03/10/deoplete.html
index b2c2cac1..4793f04a 100644
--- a/_site/2024/03/10/deoplete.html
+++ b/_site/2024/03/10/deoplete.html
@@ -166,7 +166,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/04/11/fountain-pen-nibs.html b/_site/2024/04/11/fountain-pen-nibs.html
index a3d219aa..dd66f031 100644
--- a/_site/2024/04/11/fountain-pen-nibs.html
+++ b/_site/2024/04/11/fountain-pen-nibs.html
@@ -157,7 +157,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/404.html b/_site/404.html
index e9d8549e..dd263b52 100644
--- a/_site/404.html
+++ b/_site/404.html
@@ -100,7 +100,7 @@ <h1>404</h1>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/category-theory/2023/05/18/equivalence-of-categories.html b/_site/category-theory/2023/05/18/equivalence-of-categories.html
index da672be2..ce0bfbb9 100644
--- a/_site/category-theory/2023/05/18/equivalence-of-categories.html
+++ b/_site/category-theory/2023/05/18/equivalence-of-categories.html
@@ -153,7 +153,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/category/category-theory/index.html b/_site/category/category-theory/index.html
index 835c5698..00be01ea 100644
--- a/_site/category/category-theory/index.html
+++ b/_site/category/category-theory/index.html
@@ -90,7 +90,7 @@ <h1>Archive of posts with category 'category-theory'</h1>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/category/site-construction/index.html b/_site/category/site-construction/index.html
index 05ec2cfe..242bafb1 100644
--- a/_site/category/site-construction/index.html
+++ b/_site/category/site-construction/index.html
@@ -120,7 +120,7 @@ <h1>Archive of posts with category 'site-construction'</h1>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/feed.xml b/_site/feed.xml
index 2e3204e5..56780ad1 100644
--- a/_site/feed.xml
+++ b/_site/feed.xml
@@ -1,4 +1,4 @@
-<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="4.3.3">Jekyll</generator><link href="https://hejohns.github.io/feed.xml" rel="self" type="application/atom+xml" /><link href="https://hejohns.github.io/" rel="alternate" type="text/html" /><updated>2024-06-01T16:41:42-04:00</updated><id>https://hejohns.github.io/feed.xml</id><title type="html">hejohns’ jekyll site</title><subtitle>a pure waste of space</subtitle><author><name>Johnson He</name><email>hejohns α umich δ edu</email></author><entry><title type="html">fountain pen nibs</title><link href="https://hejohns.github.io/2024/04/11/fountain-pen-nibs.html" rel="alternate" type="text/html" title="fountain pen nibs" /><published>2024-04-11T00:00:00-04:00</published><updated>2024-04-11T17:48:22-04:00</updated><id>https://hejohns.github.io/2024/04/11/fountain-pen-nibs</id><content type="html" xml:base="https://hejohns.github.io/2024/04/11/fountain-pen-nibs.html"><![CDATA[<p>Several weeks ago, I lost my other TWSBI 580 (technically, the 580 ALR)
+<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="4.3.3">Jekyll</generator><link href="https://hejohns.github.io/feed.xml" rel="self" type="application/atom+xml" /><link href="https://hejohns.github.io/" rel="alternate" type="text/html" /><updated>2024-06-03T05:14:53-04:00</updated><id>https://hejohns.github.io/feed.xml</id><title type="html">hejohns’ jekyll site</title><subtitle>a pure waste of space</subtitle><author><name>Johnson He</name><email>hejohns α umich δ edu</email></author><entry><title type="html">fountain pen nibs</title><link href="https://hejohns.github.io/2024/04/11/fountain-pen-nibs.html" rel="alternate" type="text/html" title="fountain pen nibs" /><published>2024-04-11T00:00:00-04:00</published><updated>2024-04-11T17:48:22-04:00</updated><id>https://hejohns.github.io/2024/04/11/fountain-pen-nibs</id><content type="html" xml:base="https://hejohns.github.io/2024/04/11/fountain-pen-nibs.html"><![CDATA[<p>Several weeks ago, I lost my other TWSBI 580 (technically, the 580 ALR)
 which meant that I didn’t have any EF nibs left<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup>.</p>
 
 <p>Through the process of sourcing nibs, I learned a lot about the details so:</p>
diff --git a/_site/forest.html b/_site/forest.html
index 9bb88609..105702d8 100644
--- a/_site/forest.html
+++ b/_site/forest.html
@@ -82,7 +82,7 @@ <h3>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/forest/awodey-2010.xml b/_site/forest/awodey-2010.xml
new file mode 100644
index 00000000..84da214f
--- /dev/null
+++ b/_site/forest/awodey-2010.xml
@@ -0,0 +1,10 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/blass-1988.xml b/_site/forest/blass-1988.xml
index 2a514e8e..9283abb2 100644
--- a/_site/forest/blass-1988.xml
+++ b/_site/forest/blass-1988.xml
@@ -7,14 +7,14 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>676</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -27,7 +27,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -43,60 +43,78 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>672</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
 href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>674</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
 </fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/blass-2003.xml b/_site/forest/blass-2003.xml
index e01f1eda..eeebf0fe 100644
--- a/_site/forest/blass-2003.xml
+++ b/_site/forest/blass-2003.xml
@@ -7,14 +7,14 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>667</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
diff --git a/_site/forest/carette-kiselyov-shan-2007.xml b/_site/forest/carette-kiselyov-shan-2007.xml
index 252b8239..6578707e 100644
--- a/_site/forest/carette-kiselyov-shan-2007.xml
+++ b/_site/forest/carette-kiselyov-shan-2007.xml
@@ -7,14 +7,14 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>646</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -42,7 +42,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -71,14 +71,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>645</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -99,7 +99,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
diff --git a/_site/forest/dummett-91.xml b/_site/forest/dummett-91.xml
index 2531e254..aa182b62 100644
--- a/_site/forest/dummett-91.xml
+++ b/_site/forest/dummett-91.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>508</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>717</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -17,7 +17,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/forest.json b/_site/forest/forest.json
index 1fd71c5f..4e6d4f9e 100644
--- a/_site/forest/forest.json
+++ b/_site/forest/forest.json
@@ -1 +1 @@
-{"scherer-2017":{"title":"Deciding equivalence with sums and the empty type","taxon":"Reference","tags":["scratch"],"route":"scherer-2017.xml","metas":{}},"mclarty-1992":{"title":"Elementary Categories, Elementary Toposes","taxon":"Reference","tags":[],"route":"mclarty-1992.xml","metas":{}},"friedman-1975":{"title":"Equality Between Functionals","taxon":"Reference","tags":["scratch"],"route":"friedman-1975.xml","metas":{}},"statman-1985":{"title":"Equality Between Functionals Revisited","taxon":"Reference","tags":["scratch"],"route":"statman-1985.xml","metas":{}},"hejohns-000A":{"title":"Fiberwise Structure","taxon":null,"tags":["scratch"],"route":"hejohns-000A.xml","metas":{}},"carette-kiselyov-shan-2007":{"title":"Finally Tagless, Partially Evaluated","taxon":"Reference","tags":[],"route":"carette-kiselyov-shan-2007.xml","metas":{}},"millies-zeilberger-2015":{"title":"Functors are Type Refinement 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corollary","tags":[],"route":"hejohns-0003.xml","metas":{}},"hejohns-0002":{"title":"Bisimulation","taxon":"Definition","tags":[],"route":"hejohns-0002.xml","metas":{}},"hejohns-0004":{"title":"Preservation under simulations","taxon":"Corollary","tags":[],"route":"hejohns-0004.xml","metas":{}},"hejohns-0001":{"title":"Simulation","taxon":"Definition","tags":[],"route":"hejohns-0001.xml","metas":{}},"prelude":{"title":null,"taxon":null,"tags":[],"route":"prelude.xml","metas":{}},"sub-0006":{"title":null,"taxon":"Proof","tags":[],"route":"sub-0006.xml","metas":{}},"sub-000D":{"title":null,"taxon":null,"tags":[],"route":"sub-000D.xml","metas":{}},"sub-000E":{"title":null,"taxon":"Proof corollary","tags":[],"route":"sub-000E.xml","metas":{}},"subb-0006":{"title":null,"taxon":"Proof 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+{"hejohns-000F":{"title":"\"Higher-order intuitionistic logic\" v. \"topos logic\"","taxon":null,"tags":[],"route":"hejohns-000F.xml","metas":{}},"mclarty-1990a":{"title":"Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)","taxon":"Reference","tags":[],"route":"mclarty-1990a.xml","metas":{}},"awodey-2010":{"title":"Category Theory","taxon":"Reference","tags":[],"route":"awodey-2010.xml","metas":{}},"makkai-reyes-1977":{"title":"First Order Categorical Logic","taxon":"Reference","tags":["scratch"],"route":"makkai-reyes-1977.xml","metas":{}},"lawvere-1970":{"title":"Quantifiers and Sheaves","taxon":"Reference","tags":["scratch"],"route":"lawvere-1970.xml","metas":{}},"ronnie-chen":{"title":"Ronnie Chen","taxon":"Person","tags":[],"route":"ronnie-chen.xml","metas":{}},"maclane-moerdijk-1994":{"title":"Sheaves in Geometry and Logic: A First Introduction to Topos Theory","taxon":"Reference","tags":[],"route":"maclane-moerdijk-1994.xml","metas":{}},"johnstone-2002":{"title":"Sketches of an Elephant: a topos theory compendium","taxon":"Reference","tags":[],"route":"johnstone-2002.xml","metas":{}},"mclarty-1990b":{"title":"The Uses and Abuses of the History of Topos Theory","taxon":"Reference","tags":[],"route":"mclarty-1990b.xml","metas":{}},"marquis-reyes-2012":{"title":"The history of categorical logic: 1963-1977","taxon":"Reference","tags":["scratch"],"route":"marquis-reyes-2012.xml","metas":{}},"goldblatt-2006":{"title":"Topoi: The Categorial Analysis of Logic","taxon":"Reference","tags":[],"route":"goldblatt-2006.xml","metas":{}},"hejohns-000G":{"title":"What is an elementary topos?","taxon":null,"tags":[],"route":"hejohns-000G.xml","metas":{}},"hejohns-000H":{"title":"Why is it called an \"elementary\" topos?","taxon":null,"tags":[],"route":"hejohns-000H.xml","metas":{}},"hejohns-000C":{"title":"Elementary² toposes","taxon":"Exposition","tags":["scratch"],"route":"hejohns-000C.xml","metas":{}},"hejohns-000J":{"title":"Higher-order intuitionistic topos logic","taxon":null,"tags":[],"route":"hejohns-000J.xml","metas":{}},"hejohns-000I":{"title":"Subobject classifier","taxon":"Definition","tags":["scratch"],"route":"hejohns-000I.xml","metas":{}},"scherer-2017":{"title":"Deciding equivalence with sums and the empty type","taxon":"Reference","tags":["scratch"],"route":"scherer-2017.xml","metas":{}},"mclarty-1992":{"title":"Elementary Categories, Elementary Toposes","taxon":"Reference","tags":[],"route":"mclarty-1992.xml","metas":{}},"friedman-1975":{"title":"Equality Between Functionals","taxon":"Reference","tags":["scratch"],"route":"friedman-1975.xml","metas":{}},"statman-1985":{"title":"Equality Between Functionals Revisited","taxon":"Reference","tags":["scratch"],"route":"statman-1985.xml","metas":{}},"hejohns-000A":{"title":"Fiberwise Structure","taxon":null,"tags":["scratch"],"route":"hejohns-000A.xml","metas":{}},"carette-kiselyov-shan-2007":{"title":"Finally Tagless, Partially Evaluated","taxon":"Reference","tags":[],"route":"carette-kiselyov-shan-2007.xml","metas":{}},"millies-zeilberger-2015":{"title":"Functors are Type Refinement Systems","taxon":"Reference","tags":[],"route":"millies-zeilberger-2015.xml","metas":{}},"index":{"title":"Johnson He","taxon":null,"tags":[],"route":"index.xml","metas":{"author":"False"}},"hejohns-000B":{"title":"Logical Frameworks as a meta-language","taxon":null,"tags":["scratch"],"route":"hejohns-000B.xml","metas":{}},"sterling-2022":{"title":"Naïve Logical Relations in Synthetic Tait Computability","taxon":"Reference","tags":["scratch"],"route":"sterling-2022.xml","metas":{}},"blass-2003":{"title":"Resource Consciousness in Classical Logic","taxon":"Reference","tags":["scratch"],"route":"blass-2003.xml","metas":{}},"blass-1988":{"title":"Topoi and 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diff --git a/_site/forest/friedman-1975.xml b/_site/forest/friedman-1975.xml
index 88f2dd35..2b92cf43 100644
--- a/_site/forest/friedman-1975.xml
+++ b/_site/forest/friedman-1975.xml
@@ -7,14 +7,14 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>428</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>639</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -49,7 +49,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>637</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -69,14 +69,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>638</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
diff --git a/_site/forest/goldblatt-2006.xml b/_site/forest/goldblatt-2006.xml
new file mode 100644
index 00000000..9d981387
--- /dev/null
+++ b/_site/forest/goldblatt-2006.xml
@@ -0,0 +1,17 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0001.xml b/_site/forest/hejohns-0001.xml
index b472c31a..a9e86ee0 100644
--- a/_site/forest/hejohns-0001.xml
+++ b/_site/forest/hejohns-0001.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>763</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>998</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -20,7 +20,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -40,7 +40,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -55,7 +55,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -68,7 +68,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>761</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>996</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
@@ -86,7 +86,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -99,7 +99,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -119,7 +119,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -134,7 +134,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/hejohns-0002.xml b/_site/forest/hejohns-0002.xml
index 4d62cddf..d1b88824 100644
--- a/_site/forest/hejohns-0002.xml
+++ b/_site/forest/hejohns-0002.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>755</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>990</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -27,7 +27,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -40,7 +40,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>753</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>988</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -53,7 +53,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -73,7 +73,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -88,7 +88,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/hejohns-0003.xml b/_site/forest/hejohns-0003.xml
index 7977887d..07b536de 100644
--- a/_site/forest/hejohns-0003.xml
+++ b/_site/forest/hejohns-0003.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>751</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>986</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -20,7 +20,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>749</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>984</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -40,7 +40,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/hejohns-0004.xml b/_site/forest/hejohns-0004.xml
index cb80138c..0c3f05bf 100644
--- a/_site/forest/hejohns-0004.xml
+++ b/_site/forest/hejohns-0004.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>759</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>994</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -20,7 +20,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>757</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>992</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -33,7 +33,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -53,7 +53,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -68,7 +68,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/hejohns-0005.xml b/_site/forest/hejohns-0005.xml
index cb125f1a..ad96c5ba 100644
--- a/_site/forest/hejohns-0005.xml
+++ b/_site/forest/hejohns-0005.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>681</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
@@ -25,7 +25,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -38,7 +38,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -58,7 +58,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -73,7 +73,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/hejohns-0006.xml b/_site/forest/hejohns-0006.xml
index db370f0b..081018b1 100644
--- a/_site/forest/hejohns-0006.xml
+++ b/_site/forest/hejohns-0006.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>648</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>857</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -22,7 +22,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -39,7 +39,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -58,7 +58,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -83,7 +83,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>646</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>855</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -93,7 +93,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -108,7 +108,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -125,7 +125,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -144,7 +144,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -169,7 +169,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -184,7 +184,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -212,7 +212,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -226,7 +226,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -249,7 +249,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -267,7 +267,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -305,7 +305,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -351,7 +351,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -435,7 +435,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -468,7 +468,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -499,7 +499,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -509,7 +509,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -526,7 +526,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -554,10 +554,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -572,10 +572,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -590,7 +590,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -729,7 +729,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -748,7 +748,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>640</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>849</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -773,7 +773,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>642</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>851</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -788,7 +788,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -816,7 +816,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -830,7 +830,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -853,7 +853,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -871,7 +871,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -909,7 +909,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -955,7 +955,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1039,7 +1039,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1072,7 +1072,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1103,7 +1103,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>644</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>853</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1141,7 +1141,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1187,7 +1187,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1271,7 +1271,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1304,7 +1304,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1328,7 +1328,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>639</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>848</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -1337,7 +1337,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
diff --git a/_site/forest/hejohns-0007.xml b/_site/forest/hejohns-0007.xml
index 287a5a38..d428951f 100644
--- a/_site/forest/hejohns-0007.xml
+++ b/_site/forest/hejohns-0007.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>622</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>831</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -22,7 +22,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -50,7 +50,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -64,7 +64,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -87,7 +87,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -105,7 +105,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -143,7 +143,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -189,7 +189,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -273,7 +273,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -306,7 +306,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -337,7 +337,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>829</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -347,7 +347,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -362,7 +362,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -379,7 +379,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -398,7 +398,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -423,7 +423,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -438,7 +438,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -466,7 +466,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -480,7 +480,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -503,7 +503,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -521,7 +521,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -559,7 +559,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -605,7 +605,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -689,7 +689,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -722,7 +722,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -753,7 +753,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -763,7 +763,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -780,7 +780,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -808,10 +808,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -826,10 +826,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -844,7 +844,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -983,7 +983,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1002,7 +1002,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>618</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>827</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1017,7 +1017,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1034,7 +1034,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1053,7 +1053,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1078,7 +1078,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>616</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>825</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1088,7 +1088,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -1105,7 +1105,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1133,10 +1133,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -1151,10 +1151,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -1169,7 +1169,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1308,7 +1308,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1327,7 +1327,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>824</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -1336,7 +1336,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
diff --git a/_site/forest/hejohns-0008.xml b/_site/forest/hejohns-0008.xml
index 8d2ced91..968ae28c 100644
--- a/_site/forest/hejohns-0008.xml
+++ b/_site/forest/hejohns-0008.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>673</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>882</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -17,7 +17,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -32,7 +32,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -49,7 +49,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -68,7 +68,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -93,7 +93,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -108,7 +108,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -136,7 +136,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -150,7 +150,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -173,7 +173,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -191,7 +191,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -229,7 +229,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -275,7 +275,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -359,7 +359,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -392,7 +392,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -423,7 +423,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -433,7 +433,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -450,7 +450,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -478,10 +478,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -496,10 +496,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -514,7 +514,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -653,7 +653,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -672,7 +672,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>672</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>881</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -681,7 +681,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
diff --git a/_site/forest/hejohns-0009.xml b/_site/forest/hejohns-0009.xml
index 04ccab91..e6b2bd10 100644
--- a/_site/forest/hejohns-0009.xml
+++ b/_site/forest/hejohns-0009.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>679</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -31,7 +31,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>468</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>677</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -41,7 +41,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -64,7 +64,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
diff --git a/_site/forest/hejohns-000A.xml b/_site/forest/hejohns-000A.xml
index 2f408323..029460c4 100644
--- a/_site/forest/hejohns-000A.xml
+++ b/_site/forest/hejohns-000A.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>643</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/hejohns-000B.xml b/_site/forest/hejohns-000B.xml
index 13b3a862..9f4710ef 100644
--- a/_site/forest/hejohns-000B.xml
+++ b/_site/forest/hejohns-000B.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>657</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -34,7 +34,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>445</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>656</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 href="quine-1960.xml"
diff --git a/_site/forest/hejohns-000C.xml b/_site/forest/hejohns-000C.xml
index 18143b17..b1a1bcf5 100644
--- a/_site/forest/hejohns-000C.xml
+++ b/_site/forest/hejohns-000C.xml
@@ -7,84 +7,630 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
 href="mclarty-1992.xml"
 type="local"
 addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
 </fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
 href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
     <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>455</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>458</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>460</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -97,7 +643,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
diff --git a/_site/forest/hejohns-000D.xml b/_site/forest/hejohns-000D.xml
index b00f2363..31cdaa12 100644
--- a/_site/forest/hejohns-000D.xml
+++ b/_site/forest/hejohns-000D.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>670</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>879</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -45,7 +45,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -91,7 +91,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -175,7 +175,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -208,7 +208,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -232,7 +232,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>668</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>877</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -247,7 +247,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -275,7 +275,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -289,7 +289,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -312,7 +312,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -330,7 +330,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -368,7 +368,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -414,7 +414,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -498,7 +498,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -531,7 +531,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -562,7 +562,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>660</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>869</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -578,7 +578,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>662</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>871</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -594,7 +594,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>664</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>873</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -618,7 +618,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>666</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>875</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -633,7 +633,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -650,7 +650,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -669,7 +669,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -694,7 +694,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>658</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>867</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -704,7 +704,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -721,7 +721,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -749,10 +749,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -767,10 +767,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -785,7 +785,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -924,7 +924,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/hejohns-000E.xml b/_site/forest/hejohns-000E.xml
index 602e8516..81f7de53 100644
--- a/_site/forest/hejohns-000E.xml
+++ b/_site/forest/hejohns-000E.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>637</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>846</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -17,7 +17,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -34,7 +34,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -62,10 +62,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -80,10 +80,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -98,7 +98,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -237,7 +237,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -256,7 +256,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>635</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>844</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -266,7 +266,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -281,7 +281,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -298,7 +298,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -317,7 +317,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -342,7 +342,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -357,7 +357,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -385,7 +385,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -399,7 +399,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -422,7 +422,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -440,7 +440,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -478,7 +478,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -524,7 +524,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -608,7 +608,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -641,7 +641,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -672,7 +672,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -682,7 +682,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -699,7 +699,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -727,10 +727,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -745,10 +745,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -763,7 +763,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -902,7 +902,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -921,13 +921,13 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
+root="false"><fr:frontmatter><fr:anchor>834</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
 type="local"
-addr="#277">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="#290">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -942,13 +942,13 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>627</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
+root="false"><fr:frontmatter><fr:anchor>836</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
 type="local"
-addr="#277">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="#290">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -963,7 +963,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>629</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>838</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -978,7 +978,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -1006,7 +1006,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -1020,7 +1020,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1043,7 +1043,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1061,7 +1061,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1099,7 +1099,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1145,7 +1145,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1229,7 +1229,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1262,7 +1262,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1293,7 +1293,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>631</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>840</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
 href="hejohns-0007.xml"
 type="local"
 addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -1310,7 +1310,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1333,7 +1333,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1351,7 +1351,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>633</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>842</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1389,7 +1389,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1435,7 +1435,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1519,7 +1519,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1552,7 +1552,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1576,7 +1576,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>624</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>833</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -1585,7 +1585,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
diff --git a/_site/forest/hejohns-000F.xml b/_site/forest/hejohns-000F.xml
new file mode 100644
index 00000000..86205848
--- /dev/null
+++ b/_site/forest/hejohns-000F.xml
@@ -0,0 +1,949 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title><fr:link
+href="hejohns-000C.xml"
+type="local"
+addr="hejohns-000C">Elementary² toposes</fr:link> › elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000G.xml b/_site/forest/hejohns-000G.xml
new file mode 100644
index 00000000..85a7fa78
--- /dev/null
+++ b/_site/forest/hejohns-000G.xml
@@ -0,0 +1,1256 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title><fr:link
+href="hejohns-000C.xml"
+type="local"
+addr="hejohns-000C">Elementary² toposes</fr:link> › elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+href="unstable-276.xml"
+type="local"
+addr="#276"><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000H.xml b/_site/forest/hejohns-000H.xml
new file mode 100644
index 00000000..f6795024
--- /dev/null
+++ b/_site/forest/hejohns-000H.xml
@@ -0,0 +1,1012 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title><fr:link
+href="hejohns-000C.xml"
+type="local"
+addr="hejohns-000C">Elementary² toposes</fr:link> › elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>428</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000I.xml b/_site/forest/hejohns-000I.xml
new file mode 100644
index 00000000..fb98ac29
--- /dev/null
+++ b/_site/forest/hejohns-000I.xml
@@ -0,0 +1,319 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>471</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000J.xml b/_site/forest/hejohns-000J.xml
new file mode 100644
index 00000000..86de10de
--- /dev/null
+++ b/_site/forest/hejohns-000J.xml
@@ -0,0 +1,885 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title><fr:link
+href="hejohns-000C.xml"
+type="local"
+addr="hejohns-000C">Elementary² toposes</fr:link> › elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns.xml b/_site/forest/hejohns.xml
index 7bac02b2..33ad234c 100644
--- a/_site/forest/hejohns.xml
+++ b/_site/forest/hejohns.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>746</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+root="false"><fr:frontmatter><fr:anchor>981</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
                 email: <fr:embedded-tex
 hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -16,11 +16,41 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>676</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>885</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Fibered Right Adjoints are automatically fibered?
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -28,92 +58,140 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>678</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>887</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+href="unstable-276.xml"
+type="local"
+addr="#276"><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    While reading <fr:link
-href="von-neumann-1925.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>889</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
-    "that's… the point"-- "realization" that first order logic is the
-    meta-language of axiomatic mathematics, just as (it seems) logical
-    frameworks are the meta-languages of (typically λ-calculus based)
-    programming languages. Well not literally, but the discrepancy is
-    non-substantial.
-</fr:p><fr:p>
-    That is, (classical) FoL is often the setting in which axiomatic
-    mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
-    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
-</fr:p><fr:p>
-    "definite property"
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>680</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
-href="millies-zeilberger-2015.xml"
+root="false"><fr:frontmatter><fr:anchor>891</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
 type="local"
-addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>893</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
     </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
     </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -121,1024 +199,1109 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>682</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>895</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="awodey-2010.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>684</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>897</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
-    Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
-    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
-</fr:p><fr:p>
-    Talked to Max about this on 5-9.
-    They really just give to different notions of "logic over syntax"--
-    a hoare logic, and a implicational (eg FO) logic.
-    I suppose one'd want to study a bit the relationship between the two.
-    <fr:link
-href="millies-zeilberger-2015.xml"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>686</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>899</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>901</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title><fr:link
+href="unstable-276.xml"
+type="local"
+addr="#276"><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>903</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>688</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
     </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
     </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        Johnson He
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>690</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title><fr:link
-href="dummett-91.xml"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
 type="local"
-addr="dummett-91">The Logical Basis of Metaphysics</fr:link> › Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>692</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>694</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>696</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>698</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>700</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>905</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>702</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>907</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title><fr:link
+href="hejohns-000C.xml"
+type="local"
+addr="hejohns-000C">Elementary² toposes</fr:link> › elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>704</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>706</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>708</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>710</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>712</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>714</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>716</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>718</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>720</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
-type="local"
-addr="#277">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>722</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
-type="local"
-addr="#277">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>724</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>909</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title><fr:link
+href="hejohns-000C.xml"
+type="local"
+addr="hejohns-000C">Elementary² toposes</fr:link> › elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>911</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
-type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
 type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -1146,646 +1309,618 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>726</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>728</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>913</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Fibered Right Adjoints are automatically fibered?
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>915</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    While reading <fr:link
+href="von-neumann-1925.xml"
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+    "that's… the point"-- "realization" that first order logic is the
+    meta-language of axiomatic mathematics, just as (it seems) logical
+    frameworks are the meta-languages of (typically λ-calculus based)
+    programming languages. Well not literally, but the discrepancy is
+    non-substantial.
+</fr:p><fr:p>
+    That is, (classical) FoL is often the setting in which axiomatic
+    mathematical theories are formalized. Classic (no pun intended) examples
+    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
+    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
+</fr:p><fr:p>
+    "definite property"
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>730</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
-href="hejohns-0007.xml"
+root="false"><fr:frontmatter><fr:anchor>917</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+href="millies-zeilberger-2015.xml"
 type="local"
-addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>919</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
+    Do you only actually need functors? Zeilberger's <fr:link
+href="millies-zeilberger-2015.pdf"
+type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
+</fr:p><fr:p>
+    Talked to Max about this on 5-9.
+    They really just give to different notions of "logic over syntax"--
+    a hoare logic, and a implicational (eg FO) logic.
+    I suppose one'd want to study a bit the relationship between the two.
+    <fr:link
+href="millies-zeilberger-2015.xml"
+type="local"
+addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>921</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+href="https://github.com/tomdjong/MGS-categorical-realizability"
+type="external">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>732</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
-type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
-type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
-type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>923</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
+type="external">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>925</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title><fr:link
+href="dummett-91.xml"
+type="local"
+addr="dummett-91">The Logical Basis of Metaphysics</fr:link> › Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
     </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>734</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>927</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1795,58 +1930,80 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>929</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>931</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>933</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>935</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1864,71 +2021,54 @@ type="local"
 addr="hejohns-0006">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>937</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>939</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>941</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1936,22 +2076,14 @@ hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {t
                      \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
                         X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
                      \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>943</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1960,262 +2092,55 @@ hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {t
                         A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
                         X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
                      \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>945</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>947</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
-type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>949</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
-type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
-type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
 type="local"
 addr="street-2012">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
@@ -2224,11 +2149,11 @@ addr="street-2012">"An adjunction core consists of"</fr:link>
                 a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>951</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2256,10 +2181,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -2274,10 +2199,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -2288,11 +2213,11 @@ hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {t
 </fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>953</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2427,373 +2352,632 @@ addr="hejohns-0007">the precusor</fr:link> to each
             </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>955</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
+addr="#290">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>736</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>957</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
+type="local"
+addr="#290">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>738</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>959</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>740</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>742</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>744</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="true"><fr:frontmatter><fr:anchor>675</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
-name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
+href="sub-000D.xml"
+type="local"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:p>TODO: I've copied over what's on <fr:link
-href="https://hejohns.github.io/"
-type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
-        </fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    However, unlike <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>,
+    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>961</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>963</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2808,7 +2992,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2825,7 +3009,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2844,7 +3028,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2865,54 +3049,14 @@ addr="hejohns-0006">above lemma</fr:link> is in some sense the more
     </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>965</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -2926,7 +3070,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -2949,7 +3093,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -2963,11 +3107,11 @@ hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {t
         </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>967</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -3005,7 +3149,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -3051,7 +3195,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -3135,7 +3279,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -3168,7 +3312,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -3185,155 +3329,1963 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
 href="sub-0007.xml"
 type="local"
 addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>969</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+href="sub-0006.xml"
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+        Thus <fr:link
+href="hejohns-0006.xml"
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
+addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
         The <fr:link
-href="street-2012.xml"
+href="hejohns-0006.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
+addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
+href="sub-000D.xml"
+type="local"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    However, unlike <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>,
+    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>971</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>973</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>975</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>977</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>979</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="true"><fr:frontmatter><fr:anchor>884</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
+name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
+<fr:p>TODO: I've copied over what's on <fr:link
+href="https://hejohns.github.io/"
+type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/eecs490/gradescope-utils"
+type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+        </fr:li>
+        <fr:li><fr:link
+href="https://github.com/hejohns/.rc/tree/master/home/bin"
+type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
+type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://mplse-reading-group.github.io/"
+type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+href="https://github.com/tomdjong/MGS-categorical-realizability"
+type="external">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+href="sub-0006.xml"
+type="local"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+        Thus <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
+        The <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
+href="sub-000D.xml"
+type="local"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    However, unlike <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>,
+    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
                 Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
             </fr:p>
             
@@ -3429,202 +5381,708 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="statman-1985.pdf"
+type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
 type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
+href="quine-1960.xml"
+type="local"
+addr="quine-1960">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
+href="van-heijenoort-1967.xml"
+type="local"
+addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
     </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
+        I was surprised by the mention of von Neumann 1925 in this snippet:
     </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
+type="external">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
+href="quine-1960.pdf"
+type="external">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
-    </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
-    </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
-    </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
     </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
@@ -3632,95 +6090,93 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The standard reading of type theory through the lens of category theory
-        is based on the idea of viewing a type system as a category of
-        well-typed terms. We propose a basic revision of this reading: rather
-        than interpreting type systems as categories, we describe them as
-        functors from a category of typing derivations to a category of
-        underlying terms. Then, turning this around, we explain how in fact any
-        functor gives rise to a generalized type system, with an abstract
-        notion of typing judgment, typing derivations and typing rules. This
-        leads to a purely categorical reformulation of various natural classes
-        of type systems as natural classes of functors. The main purpose of
-        this paper is to describe the general framework (which can also be seen
-        as providing a categorical analysis of refinement types), and to
-        present a few applications. As a larger case study, we revisit
-        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
-        paper’s main results may be reconstructed along these lines.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+href="maddy-2019.pdf"
+type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
     </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
     </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
@@ -3728,104 +6184,169 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
-type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
-    </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
-        <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
     </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
+    I have not read this, and have no idea what "contradictions" he's referring to.
     <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -3833,269 +6354,428 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
 </fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
-</fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
 </fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
 </fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
-    <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -4115,14 +6795,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -4157,14 +6837,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -4201,7 +6881,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -4213,7 +6893,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -4240,14 +6920,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -4268,7 +6948,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -4299,14 +6979,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -4325,7 +7005,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
 href="millies-zeilberger-2015.xml"
 type="local"
 addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -4391,70 +7071,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/index.xml b/_site/forest/index.xml
index b9e1eb4a..66f50947 100644
--- a/_site/forest/index.xml
+++ b/_site/forest/index.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="true"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
+root="true"><fr:frontmatter><fr:anchor>653</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
@@ -17,7 +17,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+root="false"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
                 email: <fr:embedded-tex
 hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
 <fr:p>TODO: I've copied over what's on <fr:link
@@ -29,7 +29,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 href="https://github.com/eecs490/gradescope-utils"
 type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
         </fr:li>
@@ -43,7 +43,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+root="false"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
 type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -52,7 +52,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 href="https://mplse-reading-group.github.io/"
 type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -61,77 +61,478 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
 href="mclarty-1992.xml"
 type="local"
 addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
 </fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
 href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
@@ -149,7 +550,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -162,7 +563,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -182,7 +583,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -197,7 +598,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -210,7 +611,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -220,7 +621,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -235,7 +636,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -252,7 +653,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -271,7 +672,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -296,7 +697,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -311,7 +712,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -339,7 +740,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -353,7 +754,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -376,7 +777,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -394,7 +795,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -432,7 +833,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -478,7 +879,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -562,7 +963,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -595,7 +996,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -626,7 +1027,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -636,7 +1037,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -653,7 +1054,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -681,10 +1082,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -699,10 +1100,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -717,7 +1118,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -856,7 +1257,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -875,60 +1276,214 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
     </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
     </fr:p><fr:p>
         Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
         with or without types, have been considered. The motivation for introducing
@@ -951,14 +1506,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -995,14 +1550,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -1030,7 +1585,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -1059,7 +1614,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1069,7 +1624,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -1092,7 +1647,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
@@ -1155,14 +1710,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -1183,7 +1738,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -1214,14 +1769,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -1240,14 +1795,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -1260,7 +1815,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -1276,7 +1831,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 href="quine-1960.xml"
@@ -1294,7 +1849,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -1303,7 +1858,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1320,7 +1875,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1386,7 +1941,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -1395,7 +1950,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -1411,14 +1966,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -1476,7 +2031,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1486,7 +2041,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1505,7 +2060,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
@@ -1515,14 +2070,588 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -1542,14 +2671,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -1584,14 +2713,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -1628,7 +2757,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1640,7 +2769,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1667,14 +2796,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -1695,7 +2824,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -1726,14 +2855,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -1752,7 +2881,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
 href="millies-zeilberger-2015.xml"
 type="local"
 addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -1818,70 +2947,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/jacobs-1999.xml b/_site/forest/jacobs-1999.xml
index 42fe8291..b0849fd1 100644
--- a/_site/forest/jacobs-1999.xml
+++ b/_site/forest/jacobs-1999.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>747</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>982</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/johnstone-2002.xml b/_site/forest/johnstone-2002.xml
new file mode 100644
index 00000000..92bef292
--- /dev/null
+++ b/_site/forest/johnstone-2002.xml
@@ -0,0 +1,10 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/lawvere-1970.pdf b/_site/forest/lawvere-1970.pdf
new file mode 100644
index 00000000..5f1a9072
Binary files /dev/null and b/_site/forest/lawvere-1970.pdf differ
diff --git a/_site/forest/lawvere-1970.xml b/_site/forest/lawvere-1970.xml
new file mode 100644
index 00000000..9219765f
--- /dev/null
+++ b/_site/forest/lawvere-1970.xml
@@ -0,0 +1,100 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/maclane-moerdijk-1994.xml b/_site/forest/maclane-moerdijk-1994.xml
new file mode 100644
index 00000000..efa985b1
--- /dev/null
+++ b/_site/forest/maclane-moerdijk-1994.xml
@@ -0,0 +1,41 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+href="unstable-276.xml"
+type="local"
+addr="#276"><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/maddy-2019.xml b/_site/forest/maddy-2019.xml
index 31c89aa2..5dd3510c 100644
--- a/_site/forest/maddy-2019.xml
+++ b/_site/forest/maddy-2019.xml
@@ -7,14 +7,14 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>503</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>712</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
diff --git a/_site/forest/makkai-reyes-1977.xml b/_site/forest/makkai-reyes-1977.xml
new file mode 100644
index 00000000..9f622b95
--- /dev/null
+++ b/_site/forest/makkai-reyes-1977.xml
@@ -0,0 +1,13 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/marquis-reyes-2012.pdf b/_site/forest/marquis-reyes-2012.pdf
new file mode 100644
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diff --git a/_site/forest/marquis-reyes-2012.xml b/_site/forest/marquis-reyes-2012.xml
new file mode 100644
index 00000000..9f08da4a
--- /dev/null
+++ b/_site/forest/marquis-reyes-2012.xml
@@ -0,0 +1,64 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>ronnie-chen</fr:addr><fr:route>ronnie-chen.xml</fr:route><fr:title>Ronnie Chen</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/mclarty-1990a.pdf b/_site/forest/mclarty-1990a.pdf
new file mode 100644
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diff --git a/_site/forest/mclarty-1990a.xml b/_site/forest/mclarty-1990a.xml
new file mode 100644
index 00000000..e9213acc
--- /dev/null
+++ b/_site/forest/mclarty-1990a.xml
@@ -0,0 +1,12 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/mclarty-1990b.pdf b/_site/forest/mclarty-1990b.pdf
new file mode 100644
index 00000000..ad70ed10
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diff --git a/_site/forest/mclarty-1990b.xml b/_site/forest/mclarty-1990b.xml
new file mode 100644
index 00000000..908033f9
--- /dev/null
+++ b/_site/forest/mclarty-1990b.xml
@@ -0,0 +1,312 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/mclarty-1992.xml b/_site/forest/mclarty-1992.xml
index fc7f793d..17b521f2 100644
--- a/_site/forest/mclarty-1992.xml
+++ b/_site/forest/mclarty-1992.xml
@@ -7,67 +7,537 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
+root="false"><fr:frontmatter><fr:anchor>636</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>631</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>633</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
 href="mclarty-1992.xml"
 type="local"
 addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
 href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>635</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
 </fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/millies-zeilberger-2015.xml b/_site/forest/millies-zeilberger-2015.xml
index bf25ff26..3eb9cbde 100644
--- a/_site/forest/millies-zeilberger-2015.xml
+++ b/_site/forest/millies-zeilberger-2015.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>649</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -17,7 +17,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -40,7 +40,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
@@ -103,7 +103,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>647</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/prelude.xml b/_site/forest/prelude.xml
index 3ab1fdb7..bad0bfe7 100644
--- a/_site/forest/prelude.xml
+++ b/_site/forest/prelude.xml
@@ -7,4 +7,4 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>765</fr:anchor><fr:addr>prelude</fr:addr><fr:route>prelude.xml</fr:route></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>1000</fr:anchor><fr:addr>prelude</fr:addr><fr:route>prelude.xml</fr:route></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/quine-1960.xml b/_site/forest/quine-1960.xml
index 3b7c21e1..690b013f 100644
--- a/_site/forest/quine-1960.xml
+++ b/_site/forest/quine-1960.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>699</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -24,7 +24,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -90,7 +90,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>489</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>698</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 href="quine-1960.xml"
@@ -108,7 +108,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>488</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>697</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 href="quine-1960.xml"
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+<g stroke='none' transform='scale(-1.00375,1.00375)translate(3.958027,-48.545919)scale(-1,-1)'>
+<g fill='#000'>
+<g stroke='none'>
+<text class='f2' x='3.958027' y='-48.545919'>Ω</text>
+</g>
+</g>
+</g>
+</g>
+</g>
+</g>
+</g>
+<g stroke-width='0.4'>
+<path d='M-19.91997 6.66507V-8.45502' fill='none'/>
+<g transform='matrix(0.0,1.0,-1.0,0.0,-19.91997,6.46507)'>
+<g stroke-dasharray='none' stroke-dashoffset='0.0'>
+ <g stroke-linecap='round'>
+ <g stroke-linejoin='round'>
+ <path d='M2.07996 2.39993C1.69995 .95996 .85316 .27998 0 0C.85316-.27998 1.69995-.95996 2.07996-2.39993' fill='none'/>
+ </g>
+ </g>
+ </g>
+</g>
+<g transform='matrix(0.0,-1.0,1.0,0.0,-19.91997,-8.65501)'>
+<g stroke-dasharray='none' stroke-dashoffset='0.0'>
+ <g stroke-linecap='round'>
+ <g stroke-linejoin='round'>
+ <path d='M-2.07996 2.39993C-1.69995 .95996-.85316 .27998 0 0C-.85316-.27998-1.69995-.95996-2.07996-2.39993' fill='none'/>
+ </g>
+ </g>
+ </g>
+</g>
+<g transform='translate(-26.04794,-1.56349)'>
+<g stroke='none' transform='scale(-1.00375,1.00375)translate(-43.902428,-48.545919)scale(-1,-1)'>
+<g fill='#000'>
+<g stroke='none'>
+<text class='f1' x='-43.902428' y='-48.545919'>𝜈</text>
+</g>
+</g>
+</g>
+</g>
+</g>
+<g stroke-width='0.4'>
+<path d='M-12.34497 15.32854H12.64998' fill='none'/>
+<g transform='translate(12.84998,15.32854)'>
+<g stroke-dasharray='none' stroke-dashoffset='0.0'>
+ <g stroke-linecap='round'>
+ <g stroke-linejoin='round'>
+ <path d='M-2.07996 2.39993C-1.69995 .95996-.85316 .27998 0 0C-.85316-.27998-1.69995-.95996-2.07996-2.39993' fill='none'/>
+ </g>
+ </g>
+ </g>
+</g>
+</g>
+<g stroke-width='0.4'>
+<g transform='translate(-11.26935,7.94781)'>
+<g stroke='none' transform='scale(-1.00375,1.00375)translate(-43.902428,-48.545919)scale(-1,-1)'>
+<g fill='#000'>
+<g stroke='none'>
+<text class='f2' x='-43.902428' y='-48.545919'>⌟</text>
+</g>
+</g>
+</g>
+</g>
+</g>
+<g stroke-width='0.4'>
+<path d='M20.05997 6.88507V-8.56503' fill='none'/>
+<g transform='matrix(0.0,1.0,-1.0,0.0,20.05997,6.68507)'>
+<g stroke-dasharray='none' stroke-dashoffset='0.0'>
+ <g stroke-linecap='round'>
+ <g stroke-linejoin='round'>
+ <path d='M2.07996 2.39993C1.69995 .95996 .85316 .27998 0 0C.85316-.27998 1.69995-.95996 2.07996-2.39993' fill='none'/>
+ </g>
+ </g>
+ </g>
+</g>
+<g transform='matrix(0.0,-1.0,1.0,0.0,20.05997,-8.76503)'>
+<g stroke-dasharray='none' stroke-dashoffset='0.0'>
+ <g stroke-linecap='round'>
+ <g stroke-linejoin='round'>
+ <path d='M-2.07996 2.39993C-1.69995 .95996-.85316 .27998 0 0C-.85316-.27998-1.69995-.95996-2.07996-2.39993' fill='none'/>
+ </g>
+ </g>
+ </g>
+</g>
+<g transform='translate(22.41496,-2.1525)'>
+<g stroke='none' transform='scale(-1.00375,1.00375)translate(-43.902428,-48.545919)scale(-1,-1)'>
+<g fill='#000'>
+<g stroke='none'>
+<text class='f1' x='-43.902428' y='-48.545919'>𝜉𝜇<tspan x='-37.123841'>𝜊𝛺</tspan></text>
+</g>
+</g>
+</g>
+</g>
+</g>
+<g stroke-width='0.4'>
+<path d='M-11.65997-17.37852H11.53998' fill='none'/>
+<g transform='translate(11.73997,-17.37852)'>
+<g stroke-dasharray='none' stroke-dashoffset='0.0'>
+ <g stroke-linecap='round'>
+ <g stroke-linejoin='round'>
+ <path d='M-2.07996 2.39993C-1.69995 .95996-.85316 .27998 0 0C-.85316-.27998-1.69995-.95996-2.07996-2.39993' fill='none'/>
+ </g>
+ </g>
+ </g>
+</g>
+<g transform='translate(-4.23,-22.82053)'>
+<g stroke='none' transform='scale(-1.00375,1.00375)translate(-43.902428,-48.545919)scale(-1,-1)'>
+<g fill='#000'>
+<g stroke='none'>
+<text class='f1' x='-43.902428' y='-48.545919'>𝝠</text>
+<text class='f0' x='-38.923101' y='-46.823375'>𝛒</text>
+</g>
+</g>
+</g>
+</g>
+</g>
+</g>
+</g>
+</g>
+</g>
+</svg>
\ No newline at end of file
diff --git a/_site/forest/ronnie-chen.xml b/_site/forest/ronnie-chen.xml
new file mode 100644
index 00000000..a4994a35
--- /dev/null
+++ b/_site/forest/ronnie-chen.xml
@@ -0,0 +1,25 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>ronnie-chen</fr:addr><fr:route>ronnie-chen.xml</fr:route><fr:title>Ronnie Chen</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/scherer-2017.xml b/_site/forest/scherer-2017.xml
index 4971b443..b134589a 100644
--- a/_site/forest/scherer-2017.xml
+++ b/_site/forest/scherer-2017.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>422</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>630</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -27,14 +27,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>628</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -69,14 +69,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>629</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
diff --git a/_site/forest/statman-1985.xml b/_site/forest/statman-1985.xml
index e08fd9e2..dfd17428 100644
--- a/_site/forest/statman-1985.xml
+++ b/_site/forest/statman-1985.xml
@@ -7,14 +7,14 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>642</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -51,7 +51,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>641</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -71,14 +71,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>640</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
diff --git a/_site/forest/sterling-2022.xml b/_site/forest/sterling-2022.xml
index f91ea7c0..a8fb4b07 100644
--- a/_site/forest/sterling-2022.xml
+++ b/_site/forest/sterling-2022.xml
@@ -7,14 +7,14 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>662</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -35,7 +35,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -66,14 +66,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>661</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -101,7 +101,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
diff --git a/_site/forest/street-2012.xml b/_site/forest/street-2012.xml
index 8d2e95fc..5068966b 100644
--- a/_site/forest/street-2012.xml
+++ b/_site/forest/street-2012.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>711</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -16,7 +16,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -32,7 +32,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>496</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>705</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -47,7 +47,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -75,7 +75,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -89,7 +89,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -112,7 +112,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -130,7 +130,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -168,7 +168,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -214,7 +214,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -298,7 +298,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -331,7 +331,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -362,7 +362,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>498</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>707</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -372,7 +372,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -389,7 +389,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -417,10 +417,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -435,10 +435,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -453,7 +453,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -592,7 +592,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -611,7 +611,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>500</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>709</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -626,7 +626,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -643,7 +643,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -662,7 +662,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/sub-0006.xml b/_site/forest/sub-0006.xml
index 404537b1..1b684136 100644
--- a/_site/forest/sub-0006.xml
+++ b/_site/forest/sub-0006.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>514</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>723</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -26,7 +26,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>721</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -41,7 +41,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -58,7 +58,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -77,7 +77,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -102,7 +102,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>719</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/sub-0007.xml b/_site/forest/sub-0007.xml
index 0f5f290a..3803a005 100644
--- a/_site/forest/sub-0007.xml
+++ b/_site/forest/sub-0007.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>656</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>865</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
 href="hejohns-0007.xml"
 type="local"
 addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -24,7 +24,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -47,7 +47,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -65,7 +65,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>654</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>863</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -80,7 +80,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -108,7 +108,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -122,7 +122,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -145,7 +145,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -163,7 +163,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -201,7 +201,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -247,7 +247,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -331,7 +331,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -364,7 +364,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -395,7 +395,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>650</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>859</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -419,7 +419,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>652</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>861</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -429,7 +429,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -446,7 +446,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -474,10 +474,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -492,10 +492,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -510,7 +510,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -649,7 +649,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/sub-000D.xml b/_site/forest/sub-000D.xml
index 5f19c12a..4da2b288 100644
--- a/_site/forest/sub-000D.xml
+++ b/_site/forest/sub-000D.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>520</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>729</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -23,7 +23,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>518</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>727</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -61,7 +61,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -107,7 +107,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -191,7 +191,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -224,7 +224,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -248,7 +248,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>516</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>725</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -286,7 +286,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -332,7 +332,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -416,7 +416,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -449,7 +449,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/sub-000E.xml b/_site/forest/sub-000E.xml
index d2837a79..94af408b 100644
--- a/_site/forest/sub-000E.xml
+++ b/_site/forest/sub-000E.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>526</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>735</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -26,7 +26,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>524</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>733</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -36,7 +36,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -53,7 +53,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -81,10 +81,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -99,10 +99,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -117,7 +117,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -256,7 +256,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -275,7 +275,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>522</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>731</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/sub-000G.xml b/_site/forest/sub-000G.xml
new file mode 100644
index 00000000..21e01425
--- /dev/null
+++ b/_site/forest/sub-000G.xml
@@ -0,0 +1,687 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+href="unstable-276.xml"
+type="local"
+addr="#276"><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/subb-0006.xml b/_site/forest/subb-0006.xml
index 2743d8e0..36d4e839 100644
--- a/_site/forest/subb-0006.xml
+++ b/_site/forest/subb-0006.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>536</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>745</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -32,7 +32,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>534</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>743</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -47,7 +47,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -64,7 +64,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -83,7 +83,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -108,7 +108,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>530</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>739</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -127,7 +127,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>532</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>741</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -142,7 +142,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -159,7 +159,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -178,7 +178,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -203,7 +203,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>528</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>737</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/subb-000D.xml b/_site/forest/subb-000D.xml
index 3c474e16..e152f211 100644
--- a/_site/forest/subb-000D.xml
+++ b/_site/forest/subb-000D.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>542</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>751</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -23,7 +23,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>540</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>749</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -61,7 +61,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -107,7 +107,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -191,7 +191,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -224,7 +224,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -248,7 +248,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>538</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>747</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -286,7 +286,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -332,7 +332,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -416,7 +416,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -449,7 +449,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/subb-000E.xml b/_site/forest/subb-000E.xml
index 179b5749..96378cb0 100644
--- a/_site/forest/subb-000E.xml
+++ b/_site/forest/subb-000E.xml
@@ -7,13 +7,13 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
+root="false"><fr:frontmatter><fr:anchor>814</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
 type="local"
-addr="#277">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="#290">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -28,7 +28,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>810</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -56,10 +56,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -74,10 +74,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -92,7 +92,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>812</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -102,7 +102,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -119,7 +119,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -147,10 +147,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -165,10 +165,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -183,7 +183,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -322,7 +322,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -341,7 +341,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>808</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -351,7 +351,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -368,7 +368,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -396,10 +396,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -414,10 +414,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -432,7 +432,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -571,7 +571,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/subbb-000D.xml b/_site/forest/subbb-000D.xml
index ee798d5a..c2522eec 100644
--- a/_site/forest/subbb-000D.xml
+++ b/_site/forest/subbb-000D.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>550</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>759</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -31,7 +31,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>548</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>757</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -69,7 +69,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -115,7 +115,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -199,7 +199,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -232,7 +232,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -256,7 +256,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>546</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>755</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
 href="hejohns-0007.xml"
 type="local"
 addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -273,7 +273,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -296,7 +296,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -314,7 +314,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>544</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>753</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -352,7 +352,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -398,7 +398,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -482,7 +482,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -515,7 +515,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/subbb-000E.xml b/_site/forest/subbb-000E.xml
index 7e9ccbe0..e9c409a0 100644
--- a/_site/forest/subbb-000E.xml
+++ b/_site/forest/subbb-000E.xml
@@ -7,13 +7,13 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
+root="false"><fr:frontmatter><fr:anchor>822</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
 type="local"
-addr="#277">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="#290">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -28,7 +28,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>818</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -56,10 +56,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -74,10 +74,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -92,7 +92,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>820</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -102,7 +102,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -119,7 +119,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -147,10 +147,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -165,10 +165,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -183,7 +183,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -322,7 +322,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -341,7 +341,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>816</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -351,7 +351,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -368,7 +368,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -396,10 +396,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -414,10 +414,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -432,7 +432,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -571,7 +571,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/subsub-0007.xml b/_site/forest/subsub-0007.xml
index 350e36d3..7896e463 100644
--- a/_site/forest/subsub-0007.xml
+++ b/_site/forest/subsub-0007.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>556</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>765</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -22,7 +22,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>552</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>761</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
 href="hejohns-0007.xml"
 type="local"
 addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -39,7 +39,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -62,7 +62,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -80,7 +80,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>554</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>763</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -118,7 +118,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -164,7 +164,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -248,7 +248,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -281,7 +281,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/subsubb-0007.xml b/_site/forest/subsubb-0007.xml
index 1cf377e6..29db181c 100644
--- a/_site/forest/subsubb-0007.xml
+++ b/_site/forest/subsubb-0007.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>560</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>769</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -23,7 +23,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>558</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>767</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
 href="hejohns-0007.xml"
 type="local"
 addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -40,7 +40,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -63,7 +63,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
diff --git a/_site/forest/unstable-253.xml b/_site/forest/unstable-253.xml
index 6c33361d..9016b3da 100644
--- a/_site/forest/unstable-253.xml
+++ b/_site/forest/unstable-253.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>493</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>702</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -23,7 +23,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>492</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>701</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -32,7 +32,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
diff --git a/_site/forest/unstable-254.xml b/_site/forest/unstable-254.xml
index be52acc7..13d461a2 100644
--- a/_site/forest/unstable-254.xml
+++ b/_site/forest/unstable-254.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>479</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -28,14 +28,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>478</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -56,7 +56,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
diff --git a/_site/forest/unstable-255.xml b/_site/forest/unstable-255.xml
index c494d36c..301613ee 100644
--- a/_site/forest/unstable-255.xml
+++ b/_site/forest/unstable-255.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>484</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -38,14 +38,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>483</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -66,7 +66,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -97,14 +97,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>482</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -132,7 +132,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
diff --git a/_site/forest/unstable-256.xml b/_site/forest/unstable-256.xml
index d29e8a2a..25255e13 100644
--- a/_site/forest/unstable-256.xml
+++ b/_site/forest/unstable-256.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>489</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -42,14 +42,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>488</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -86,14 +86,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>486</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
diff --git a/_site/forest/unstable-257.xml b/_site/forest/unstable-257.xml
index 3ff0df59..3cf3ca60 100644
--- a/_site/forest/unstable-257.xml
+++ b/_site/forest/unstable-257.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>685</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -67,7 +67,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>683</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -84,7 +84,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/unstable-258.xml b/_site/forest/unstable-258.xml
index 8a4a7dae..dc6c8121 100644
--- a/_site/forest/unstable-258.xml
+++ b/_site/forest/unstable-258.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>495</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -30,7 +30,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>493</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -40,7 +40,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -63,7 +63,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
diff --git a/_site/forest/unstable-259.xml b/_site/forest/unstable-259.xml
index 46462bb2..1723d937 100644
--- a/_site/forest/unstable-259.xml
+++ b/_site/forest/unstable-259.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>670</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
 href="millies-zeilberger-2015.xml"
 type="local"
 addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -73,7 +73,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>668</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -83,7 +83,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -106,7 +106,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
diff --git a/_site/forest/unstable-260.xml b/_site/forest/unstable-260.xml
index d86a0cc0..ee9e6144 100644
--- a/_site/forest/unstable-260.xml
+++ b/_site/forest/unstable-260.xml
@@ -7,22 +7,17 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>495</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
+root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
     </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -30,62 +25,24 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>494</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
-</fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
-</fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
-</fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
-    <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-261.xml b/_site/forest/unstable-261.xml
index db38037a..684e6bb8 100644
--- a/_site/forest/unstable-261.xml
+++ b/_site/forest/unstable-261.xml
@@ -7,1435 +7,37 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title><fr:link
-href="index.xml"
-type="local"
-addr="index">Johnson He</fr:link> › Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
-        </fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="true"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
-name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:p>TODO: I've copied over what's on <fr:link
-href="https://hejohns.github.io/"
-type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
-        </fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
-type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
-type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
-type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
-            <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
-                </fr:li>
-                <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
-    </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
-    </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
-    </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
-    </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
-    </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The standard reading of type theory through the lens of category theory
-        is based on the idea of viewing a type system as a category of
-        well-typed terms. We propose a basic revision of this reading: rather
-        than interpreting type systems as categories, we describe them as
-        functors from a category of typing derivations to a category of
-        underlying terms. Then, turning this around, we explain how in fact any
-        functor gives rise to a generalized type system, with an abstract
-        notion of typing judgment, typing derivations and typing rules. This
-        leads to a purely categorical reformulation of various natural classes
-        of type systems as natural classes of functors. The main purpose of
-        this paper is to describe the general framework (which can also be seen
-        as providing a categorical analysis of refinement types), and to
-        present a few applications. As a larger case study, we revisit
-        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
-        paper’s main results may be reconstructed along these lines.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
-    </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
-type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
-    </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
-        <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>704</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
-type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
-</fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
-type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
-    </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>703</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -1486,433 +88,4 @@ root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:ta
     I've used quotes here as air quotes, not actual quotes of Maddy.
     <fr:link
 href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Fibered Right Adjoints are automatically fibered?
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    While reading <fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
-    "that's… the point"-- "realization" that first order logic is the
-    meta-language of axiomatic mathematics, just as (it seems) logical
-    frameworks are the meta-languages of (typically λ-calculus based)
-    programming languages. Well not literally, but the discrepancy is
-    non-substantial.
-</fr:p><fr:p>
-    That is, (classical) FoL is often the setting in which axiomatic
-    mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
-    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
-</fr:p><fr:p>
-    "definite property"
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
-type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
-    </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
-        <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
-href="millies-zeilberger-2015.xml"
-type="local"
-addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
-    </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
-    Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
-    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
-</fr:p><fr:p>
-    Talked to Max about this on 5-9.
-    They really just give to different notions of "logic over syntax"--
-    a hoare logic, and a implicational (eg FO) logic.
-    I suppose one'd want to study a bit the relationship between the two.
-    <fr:link
-href="millies-zeilberger-2015.xml"
-type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-262.xml b/_site/forest/unstable-262.xml
index 6f30de2d..7b768518 100644
--- a/_site/forest/unstable-262.xml
+++ b/_site/forest/unstable-262.xml
@@ -7,1907 +7,102 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>449</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title><fr:link
-href="index.xml"
-type="local"
-addr="index">Johnson He</fr:link> › Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="true"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
-name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:p>TODO: I've copied over what's on <fr:link
-href="https://hejohns.github.io/"
-type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
-        </fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
-type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
-type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
-type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
-            <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
-                </fr:li>
-                <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
+root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
     </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
-    </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
-    </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
-    </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
-    </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
-    </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The standard reading of type theory through the lens of category theory
-        is based on the idea of viewing a type system as a category of
-        well-typed terms. We propose a basic revision of this reading: rather
-        than interpreting type systems as categories, we describe them as
-        functors from a category of typing derivations to a category of
-        underlying terms. Then, turning this around, we explain how in fact any
-        functor gives rise to a generalized type system, with an abstract
-        notion of typing judgment, typing derivations and typing rules. This
-        leads to a purely categorical reformulation of various natural classes
-        of type systems as natural classes of functors. The main purpose of
-        this paper is to describe the general framework (which can also be seen
-        as providing a categorical analysis of refinement types), and to
-        present a few applications. As a larger case study, we revisit
-        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
-        paper’s main results may be reconstructed along these lines.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
-    </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
-type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
-    </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
-        <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
-type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
-</fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
-type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
-    </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
-</fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
-</fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
-</fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
-    <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Fibered Right Adjoints are automatically fibered?
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    While reading <fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
-    "that's… the point"-- "realization" that first order logic is the
-    meta-language of axiomatic mathematics, just as (it seems) logical
-    frameworks are the meta-languages of (typically λ-calculus based)
-    programming languages. Well not literally, but the discrepancy is
-    non-substantial.
-</fr:p><fr:p>
-    That is, (classical) FoL is often the setting in which axiomatic
-    mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
-    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
-</fr:p><fr:p>
-    "definite property"
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
-type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
-    </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
-        <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
-href="millies-zeilberger-2015.xml"
-type="local"
-addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
-    </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
-    Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
-    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
-</fr:p><fr:p>
-    Talked to Max about this on 5-9.
-    They really just give to different notions of "logic over syntax"--
-    a hoare logic, and a implicational (eg FO) logic.
-    I suppose one'd want to study a bit the relationship between the two.
+    I have not read this, and have no idea what "contradictions" he's referring to.
     <fr:link
-href="millies-zeilberger-2015.xml"
-type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-263.xml b/_site/forest/unstable-263.xml
index 6a53656b..60f06eff 100644
--- a/_site/forest/unstable-263.xml
+++ b/_site/forest/unstable-263.xml
@@ -7,19 +7,24 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>627</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title><fr:link
 href="index.xml"
 type="local"
-addr="index">Johnson He</fr:link> › Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+addr="index">Johnson He</fr:link> › Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/eecs490/gradescope-utils"
+type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+        </fr:li>
+        <fr:li><fr:link
+href="https://github.com/hejohns/.rc/tree/master/home/bin"
+type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="true"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
+root="true"><fr:frontmatter><fr:anchor>626</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
@@ -29,7 +34,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+root="false"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
                 email: <fr:embedded-tex
 hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
 <fr:p>TODO: I've copied over what's on <fr:link
@@ -41,7 +46,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 href="https://github.com/eecs490/gradescope-utils"
 type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
         </fr:li>
@@ -55,7 +60,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+root="false"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
 type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -64,7 +69,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 href="https://mplse-reading-group.github.io/"
 type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -73,77 +78,478 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
 href="mclarty-1992.xml"
 type="local"
 addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
 </fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
 href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
@@ -161,7 +567,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -174,7 +580,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -194,7 +600,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -209,7 +615,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -222,7 +628,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -232,7 +638,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -247,7 +653,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -264,7 +670,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -283,7 +689,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -308,7 +714,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -323,7 +729,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -351,7 +757,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -365,7 +771,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -388,7 +794,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -406,7 +812,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -444,7 +850,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -490,7 +896,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -574,7 +980,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -607,7 +1013,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -638,7 +1044,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -648,7 +1054,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -665,7 +1071,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -693,10 +1099,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -711,10 +1117,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -729,7 +1135,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -868,7 +1274,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -887,52 +1293,206 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
         embryonic state of development, awaiting further conceptual advances.
     </fr:p><fr:p>
         There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
@@ -963,14 +1523,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -1007,14 +1567,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -1042,7 +1602,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -1071,7 +1631,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1081,7 +1641,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -1104,7 +1664,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
@@ -1167,14 +1727,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -1195,7 +1755,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -1226,14 +1786,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -1252,14 +1812,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -1272,7 +1832,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -1288,7 +1848,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 href="quine-1960.xml"
@@ -1306,7 +1866,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -1315,7 +1875,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1332,7 +1892,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1398,7 +1958,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -1407,7 +1967,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -1423,14 +1983,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -1488,7 +2048,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1498,7 +2058,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1517,7 +2077,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
@@ -1527,14 +2087,588 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -1554,14 +2688,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -1596,14 +2730,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -1640,7 +2774,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1652,7 +2786,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1679,14 +2813,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -1707,7 +2841,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -1738,14 +2872,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -1764,7 +2898,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
 href="millies-zeilberger-2015.xml"
 type="local"
 addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -1830,70 +2964,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/unstable-264.xml b/_site/forest/unstable-264.xml
index 679fc59c..f15fa74c 100644
--- a/_site/forest/unstable-264.xml
+++ b/_site/forest/unstable-264.xml
@@ -7,159 +7,80 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>660</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title><fr:link
 href="index.xml"
 type="local"
-addr="index">Johnson He</fr:link> › Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+addr="index">Johnson He</fr:link> › Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
+type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+expanded="true"
+root="true"><fr:frontmatter><fr:anchor>659</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
+name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
+<fr:p>TODO: I've copied over what's on <fr:link
+href="https://hejohns.github.io/"
+type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
+root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/eecs490/gradescope-utils"
+type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+        </fr:li>
+        <fr:li><fr:link
+href="https://github.com/hejohns/.rc/tree/master/home/bin"
+type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
+type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://mplse-reading-group.github.io/"
+type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -169,31 +90,59 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -201,58 +150,72 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -260,314 +223,299 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
-</fr:p><fr:tree
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -575,545 +523,193 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+href="https://github.com/tomdjong/MGS-categorical-realizability"
+type="external">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
-            <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
-                </fr:li>
-                <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="true"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
-name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:p>TODO: I've copied over what's on <fr:link
-href="https://hejohns.github.io/"
-type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
-        </fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+href="sub-0006.xml"
+type="local"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+        Thus <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
+        The <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1128,7 +724,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -1156,7 +752,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -1170,7 +766,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1193,7 +789,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1211,7 +807,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1249,7 +845,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1295,7 +891,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1379,7 +975,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1412,7 +1008,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1443,7 +1039,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1453,7 +1049,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -1470,7 +1066,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1498,10 +1094,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -1516,10 +1112,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -1534,7 +1130,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1673,7 +1269,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1692,93 +1288,247 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
     </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
     </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
+    I have not read this, and have no idea what "contradictions" he's referring to.
     <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
 type="local"
 addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
         theory of the typed λ-calculus. In order to do this, it is appropriate to
@@ -1812,14 +1562,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -1847,7 +1597,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -1864,44 +1614,627 @@ root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon
         A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
         the semantics of the DSL/embedded/object language's numeric type.
     </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
+type="local"
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
+href="quine-1960.xml"
+type="local"
+addr="quine-1960">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
+href="van-heijenoort-1967.xml"
+type="local"
+addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
+type="external">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
+href="quine-1960.pdf"
+type="external">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+href="maddy-2019.pdf"
+type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The standard reading of type theory through the lens of category theory
-        is based on the idea of viewing a type system as a category of
-        well-typed terms. We propose a basic revision of this reading: rather
-        than interpreting type systems as categories, we describe them as
-        functors from a category of typing derivations to a category of
-        underlying terms. Then, turning this around, we explain how in fact any
-        functor gives rise to a generalized type system, with an abstract
-        notion of typing judgment, typing derivations and typing rules. This
-        leads to a purely categorical reformulation of various natural classes
-        of type systems as natural classes of functors. The main purpose of
-        this paper is to describe the general framework (which can also be seen
-        as providing a categorical analysis of refinement types), and to
-        present a few applications. As a larger case study, we revisit
-        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
-        paper’s main results may be reconstructed along these lines.
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -1909,167 +2242,138 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
-    </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
-    </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
-        <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -2077,269 +2381,289 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
-</fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
-</fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
 </fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
 </fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
-    <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -2359,14 +2683,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -2401,14 +2725,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -2445,7 +2769,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2457,7 +2781,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2484,14 +2808,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -2512,7 +2836,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -2543,14 +2867,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -2569,7 +2893,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
 href="millies-zeilberger-2015.xml"
 type="local"
 addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -2635,70 +2959,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/unstable-265.xml b/_site/forest/unstable-265.xml
index cb447ee5..08394035 100644
--- a/_site/forest/unstable-265.xml
+++ b/_site/forest/unstable-265.xml
@@ -7,219 +7,281 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>455</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>655</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title><fr:link
 href="index.xml"
 type="local"
-addr="index">Johnson He</fr:link> › References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+addr="index">Johnson He</fr:link> › Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://mplse-reading-group.github.io/"
+type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+expanded="true"
+root="true"><fr:frontmatter><fr:anchor>654</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
+name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
+<fr:p>TODO: I've copied over what's on <fr:link
+href="https://hejohns.github.io/"
+type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/eecs490/gradescope-utils"
+type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+        </fr:li>
+        <fr:li><fr:link
+href="https://github.com/hejohns/.rc/tree/master/home/bin"
+type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
+type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://mplse-reading-group.github.io/"
+type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
+        Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
-                completness theorems,
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
             </fr:li>
             <fr:li>
-                the solvability of higher type functional equations, and
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
             </fr:li>
             <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
-    </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
-    </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
-    </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
-    </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
-    </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The standard reading of type theory through the lens of category theory
-        is based on the idea of viewing a type system as a category of
-        well-typed terms. We propose a basic revision of this reading: rather
-        than interpreting type systems as categories, we describe them as
-        functors from a category of typing derivations to a category of
-        underlying terms. Then, turning this around, we explain how in fact any
-        functor gives rise to a generalized type system, with an abstract
-        notion of typing judgment, typing derivations and typing rules. This
-        leads to a purely categorical reformulation of various natural classes
-        of type systems as natural classes of functors. The main purpose of
-        this paper is to describe the general framework (which can also be seen
-        as providing a categorical analysis of refinement types), and to
-        present a few applications. As a larger case study, we revisit
-        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
-        paper’s main results may be reconstructed along these lines.
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -227,90 +289,60 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
-    </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -318,211 +350,259 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
-        <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
 </fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
+    And <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+href="https://github.com/tomdjong/MGS-categorical-realizability"
+type="external">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
 </fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
-    </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -530,473 +610,163 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
-</fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
-</fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
-</fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
-    <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
     </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="true"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
-name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:p>TODO: I've copied over what's on <fr:link
-href="https://hejohns.github.io/"
-type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+href="sub-0006.xml"
+type="local"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+        Thus <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
+        The <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
-        </fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1019,7 +789,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1037,7 +807,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1075,7 +845,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1121,7 +891,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1205,7 +975,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -1238,7 +1008,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1269,7 +1039,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1279,7 +1049,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -1296,7 +1066,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1324,10 +1094,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -1342,10 +1112,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -1360,7 +1130,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1499,7 +1269,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1518,21 +1288,175 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
 type="local"
 addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
     Referenced on <fr:link
@@ -1545,21 +1469,21 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -1594,14 +1518,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -1638,14 +1562,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -1673,7 +1597,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -1702,7 +1626,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1712,7 +1636,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -1735,7 +1659,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
@@ -1798,14 +1722,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -1826,7 +1750,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -1857,14 +1781,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -1883,14 +1807,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -1903,7 +1827,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -1919,7 +1843,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 href="quine-1960.xml"
@@ -1937,7 +1861,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -1946,7 +1870,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1963,209 +1887,783 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
+type="external">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
+href="quine-1960.pdf"
+type="external">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+href="maddy-2019.pdf"
+type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
-</fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
 </fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
 </fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
-    <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -2185,14 +2683,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -2227,14 +2725,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -2271,7 +2769,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2283,7 +2781,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2310,14 +2808,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -2338,7 +2836,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -2369,14 +2867,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -2395,7 +2893,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
 href="millies-zeilberger-2015.xml"
 type="local"
 addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -2461,70 +2959,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/unstable-266.xml b/_site/forest/unstable-266.xml
index 4620f248..3c5f9258 100644
--- a/_site/forest/unstable-266.xml
+++ b/_site/forest/unstable-266.xml
@@ -7,182 +7,168 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>664</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title><fr:link
 href="index.xml"
 type="local"
-addr="index">Johnson He</fr:link> › Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+addr="index">Johnson He</fr:link> › Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
+        Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
-                completness theorems,
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
             </fr:li>
             <fr:li>
-                the solvability of higher type functional equations, and
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
             </fr:li>
             <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Fibered Right Adjoints are automatically fibered?
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    While reading <fr:link
-href="von-neumann-1925.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
-    "that's… the point"-- "realization" that first order logic is the
-    meta-language of axiomatic mathematics, just as (it seems) logical
-    frameworks are the meta-languages of (typically λ-calculus based)
-    programming languages. Well not literally, but the discrepancy is
-    non-substantial.
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
 </fr:p><fr:p>
-    That is, (classical) FoL is often the setting in which axiomatic
-    mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
-    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
-    "definite property"
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -190,341 +176,312 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
-href="millies-zeilberger-2015.xml"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
-    Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
-    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
-</fr:p><fr:p>
-    Talked to Max about this on 5-9.
-    They really just give to different notions of "logic over syntax"--
-    a hoare logic, and a implicational (eg FO) logic.
-    I suppose one'd want to study a bit the relationship between the two.
-    <fr:link
-href="millies-zeilberger-2015.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="true"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
-name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:p>TODO: I've copied over what's on <fr:link
-href="https://hejohns.github.io/"
-type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
-        </fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
@@ -542,7 +499,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -555,7 +512,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -575,7 +532,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -590,7 +547,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -603,7 +560,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -613,7 +570,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -628,7 +585,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -645,7 +602,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -664,7 +621,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -689,7 +646,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -704,7 +661,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -732,7 +689,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
@@ -746,7 +703,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -769,7 +726,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -787,7 +744,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -825,7 +782,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -871,7 +828,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -955,7 +912,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
@@ -988,7 +945,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1019,7 +976,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -1029,7 +986,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
@@ -1046,7 +1003,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1074,10 +1031,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -1092,10 +1049,10 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
 hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
              \begin {tikzcd}
@@ -1110,7 +1067,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1249,7 +1206,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -1261,161 +1218,276 @@ href="subb-0006.xml"
 type="local"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="true"><fr:frontmatter><fr:anchor>663</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
+name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
+<fr:p>TODO: I've copied over what's on <fr:link
+href="https://hejohns.github.io/"
+type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/eecs490/gradescope-utils"
+type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+        </fr:li>
+        <fr:li><fr:link
+href="https://github.com/hejohns/.rc/tree/master/home/bin"
+type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
+type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://mplse-reading-group.github.io/"
+type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
+        Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
-                completness theorems,
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
             </fr:li>
             <fr:li>
-                the solvability of higher type functional equations, and
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
             </fr:li>
             <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
-    </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -1423,61 +1495,1952 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
     </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
     </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+href="https://github.com/tomdjong/MGS-categorical-realizability"
+type="external">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+href="sub-0006.xml"
+type="local"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+        Thus <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
+        The <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
+href="sub-000D.xml"
+type="local"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    However, unlike <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>,
+    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="statman-1985.pdf"
+type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
+type="local"
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
+href="quine-1960.xml"
+type="local"
+addr="quine-1960">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
+href="van-heijenoort-1967.xml"
+type="local"
+addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
+type="external">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
+href="quine-1960.pdf"
+type="external">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+href="maddy-2019.pdf"
+type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The standard reading of type theory through the lens of category theory
-        is based on the idea of viewing a type system as a category of
-        well-typed terms. We propose a basic revision of this reading: rather
-        than interpreting type systems as categories, we describe them as
-        functors from a category of typing derivations to a category of
-        underlying terms. Then, turning this around, we explain how in fact any
-        functor gives rise to a generalized type system, with an abstract
-        notion of typing judgment, typing derivations and typing rules. This
-        leads to a purely categorical reformulation of various natural classes
-        of type systems as natural classes of functors. The main purpose of
-        this paper is to describe the general framework (which can also be seen
-        as providing a categorical analysis of refinement types), and to
-        present a few applications. As a larger case study, we revisit
-        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
-        paper’s main results may be reconstructed along these lines.
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -1485,90 +3448,88 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
-    </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -1576,76 +3537,110 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -1653,134 +3648,100 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
-</fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
-    </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -1788,134 +3749,127 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
 </fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
 </fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
 </fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
-    <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="https://arxiv.org/pdf/1610.01213"
 type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -1935,14 +3889,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -1977,14 +3931,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 href="friedman-1975.xml"
@@ -2021,7 +3975,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2033,7 +3987,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -2060,14 +4014,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -2088,7 +4042,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -2119,14 +4073,14 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -2145,7 +4099,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
 href="millies-zeilberger-2015.xml"
 type="local"
 addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
@@ -2211,70 +4165,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The following are some basic (alt. elementary) questions, whose "elevator
-    pitch" answers ought to roll off the tongue:
-    <fr:ul><fr:li>
-            What is a(n elementary) topos?
-        </fr:li>
-        <fr:li>
-            What is this "topos logic" that is sound in any elementary topos?
-            As in literally, what is the syntax?
-            Is it the same thing as this "higher-order intuitionistic logic"?
-            What are the inference rules/axioms?
-        </fr:li>
-        <fr:li>
-            What is this "Kripke-Joyal semantics"?
-            Is this the same thing as the so-called "forcing semantics"?
-        </fr:li>
-        <fr:li>
-            What is this "Mitchell-Bénabou language"?
-            Is it the same thing as "topos logic"?
-        </fr:li></fr:ul>
-    Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-</fr:p><fr:p>
-    Some more difficult questions we do not attempt to answer at the moment:
-    <fr:ul><fr:li>
-            What is a Grothendieck topos, and what's the difference v.
-            elementary toposes?
-        </fr:li>
-        <fr:li>
-            What is a "geometric logic" and a "geometric morphism"?
-            And what's the difference v. a "logical morphism"?
-        </fr:li>
-        <fr:li>
-            In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-            (What is forcing??)
-        </fr:li>
-        <fr:li>
-            And everything else…
-        </fr:li></fr:ul></fr:p><fr:p>
-    Many object to the name "higher-order intuitionistic logic".
-    <fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> says "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic."
-    <fr:link
-href="maclarty-1992"
-type="external">maclarty-1992</fr:link> says "TODO".
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/forest/unstable-267.xml b/_site/forest/unstable-267.xml
index a2486d62..b66a1ecd 100644
--- a/_site/forest/unstable-267.xml
+++ b/_site/forest/unstable-267.xml
@@ -7,318 +7,3320 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>564</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>666</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title><fr:link
+href="index.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+addr="index">Johnson He</fr:link> › References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>562</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
+addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="statman-1985.pdf"
+type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
-type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
+href="quine-1960.xml"
+type="local"
+addr="quine-1960">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
+href="van-heijenoort-1967.xml"
+type="local"
+addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
+type="external">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
+href="quine-1960.pdf"
+type="external">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+href="maddy-2019.pdf"
+type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="true"><fr:frontmatter><fr:anchor>665</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
+name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
+<fr:p>TODO: I've copied over what's on <fr:link
+href="https://hejohns.github.io/"
+type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/eecs490/gradescope-utils"
+type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+        </fr:li>
+        <fr:li><fr:link
+href="https://github.com/hejohns/.rc/tree/master/home/bin"
+type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
+type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://mplse-reading-group.github.io/"
+type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+href="https://github.com/tomdjong/MGS-categorical-realizability"
+type="external">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+href="sub-0006.xml"
+type="local"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+        Thus <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
+        The <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
 display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
             <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
 href="sub-000D.xml"
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    However, unlike <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>,
+    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="statman-1985.pdf"
+type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
+type="local"
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
+href="quine-1960.xml"
+type="local"
+addr="quine-1960">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
+href="van-heijenoort-1967.xml"
+type="local"
+addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
+type="external">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
+href="quine-1960.pdf"
+type="external">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+href="maddy-2019.pdf"
+type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -326,28 +3328,437 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
     </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
+addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="statman-1985.pdf"
+type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Fibered Right Adjoints are automatically fibered?
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    While reading <fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+    "that's… the point"-- "realization" that first order logic is the
+    meta-language of axiomatic mathematics, just as (it seems) logical
+    frameworks are the meta-languages of (typically λ-calculus based)
+    programming languages. Well not literally, but the discrepancy is
+    non-substantial.
+</fr:p><fr:p>
+    That is, (classical) FoL is often the setting in which axiomatic
+    mathematical theories are formalized. Classic (no pun intended) examples
+    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
+    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
+</fr:p><fr:p>
+    "definite property"
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
+type="local"
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+href="millies-zeilberger-2015.xml"
+type="local"
+addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
+    Do you only actually need functors? Zeilberger's <fr:link
+href="millies-zeilberger-2015.pdf"
+type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
+</fr:p><fr:p>
+    Talked to Max about this on 5-9.
+    They really just give to different notions of "logic over syntax"--
+    a hoare logic, and a implicational (eg FO) logic.
+    I suppose one'd want to study a bit the relationship between the two.
+    <fr:link
+href="millies-zeilberger-2015.xml"
+type="local"
+addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-268.xml b/_site/forest/unstable-268.xml
index e4951947..7153b957 100644
--- a/_site/forest/unstable-268.xml
+++ b/_site/forest/unstable-268.xml
@@ -7,31 +7,3199 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>568</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>652</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title><fr:link
+href="index.xml"
+type="local"
+addr="index">Johnson He</fr:link> › Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="statman-1985.pdf"
+type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Fibered Right Adjoints are automatically fibered?
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    While reading <fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+    "that's… the point"-- "realization" that first order logic is the
+    meta-language of axiomatic mathematics, just as (it seems) logical
+    frameworks are the meta-languages of (typically λ-calculus based)
+    programming languages. Well not literally, but the discrepancy is
+    non-substantial.
+</fr:p><fr:p>
+    That is, (classical) FoL is often the setting in which axiomatic
+    mathematical theories are formalized. Classic (no pun intended) examples
+    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
+    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
+</fr:p><fr:p>
+    "definite property"
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
+type="local"
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+href="millies-zeilberger-2015.xml"
+type="local"
+addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
+    Do you only actually need functors? Zeilberger's <fr:link
+href="millies-zeilberger-2015.pdf"
+type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
+</fr:p><fr:p>
+    Talked to Max about this on 5-9.
+    They really just give to different notions of "logic over syntax"--
+    a hoare logic, and a implicational (eg FO) logic.
+    I suppose one'd want to study a bit the relationship between the two.
+    <fr:link
+href="millies-zeilberger-2015.xml"
+type="local"
+addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="true"><fr:frontmatter><fr:anchor>651</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:meta
+name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>9</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>My screen name is <fr:code>hejohns</fr:code>. This is my forest.</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li></fr:ul></fr:mainmatter></fr:tree>
+<fr:p>TODO: I've copied over what's on <fr:link
+href="https://hejohns.github.io/"
+type="external">https://hejohns.github.io/</fr:link>, but this isn't a good selection.</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>#263</fr:addr><fr:route>unstable-263.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/eecs490/gradescope-utils"
+type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+        </fr:li>
+        <fr:li><fr:link
+href="https://github.com/hejohns/.rc/tree/master/home/bin"
+type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#264</fr:addr><fr:route>unstable-264.xml</fr:route><fr:title>Media</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
+type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>#265</fr:addr><fr:route>unstable-265.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+href="https://mplse-reading-group.github.io/"
+type="external">Placeholder Link</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:addr>#266</fr:addr><fr:route>unstable-266.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+href="https://github.com/tomdjong/MGS-categorical-realizability"
+type="external">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>567</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+href="sub-0006.xml"
+type="local"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+        Thus <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
+        The <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
+href="sub-000D.xml"
+type="local"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    However, unlike <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>,
+    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="mclarty-1990a.pdf"
+type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="statman-1985.pdf"
+type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>583</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#258</fr:addr><fr:route>unstable-258.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
+type="local"
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
+href="quine-1960.xml"
+type="local"
+addr="quine-1960">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
+href="van-heijenoort-1967.xml"
+type="local"
+addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
+type="external">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
+href="quine-1960.pdf"
+type="external">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>596</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>598</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#261</fr:addr><fr:route>unstable-261.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+href="maddy-2019.pdf"
+type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>566</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -39,16 +3207,65 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#268</fr:addr><fr:route>unstable-268.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -56,18 +3273,34 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
     </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -75,22 +3308,564 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="https://arxiv.org/pdf/1610.01213"
+type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
+href="statman-1985.xml"
+type="local"
+addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+href="https://proofassistants.stackexchange.com/a/1177"
+type="external">stack overflow</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>613</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#256</fr:addr><fr:route>unstable-256.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+href="friedman-1975.xml"
+type="local"
+addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="statman-1985.pdf"
+type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Fibered Right Adjoints are automatically fibered?
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    While reading <fr:link
+href="von-neumann-1925.xml"
+type="local"
+addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+    "that's… the point"-- "realization" that first order logic is the
+    meta-language of axiomatic mathematics, just as (it seems) logical
+    frameworks are the meta-languages of (typically λ-calculus based)
+    programming languages. Well not literally, but the discrepancy is
+    non-substantial.
+</fr:p><fr:p>
+    That is, (classical) FoL is often the setting in which axiomatic
+    mathematical theories are formalized. Classic (no pun intended) examples
+    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
+    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
+</fr:p><fr:p>
+    "definite property"
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#254</fr:addr><fr:route>unstable-254.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>#255</fr:addr><fr:route>unstable-255.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
+href="carette-kiselyov-shan-2007.xml"
+type="local"
+addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+        and <fr:link
+href="hoffman-1999"
+type="external">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+href="sterling-2022.pdf"
+type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>#259</fr:addr><fr:route>unstable-259.xml</fr:route><fr:title><fr:link
+href="millies-zeilberger-2015.xml"
+type="local"
+addr="millies-zeilberger-2015">Functors are Type Refinement Systems</fr:link> › Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
+    Do you only actually need functors? Zeilberger's <fr:link
+href="millies-zeilberger-2015.pdf"
+type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
+</fr:p><fr:p>
+    Talked to Max about this on 5-9.
+    They really just give to different notions of "logic over syntax"--
+    a hoare logic, and a implicational (eg FO) logic.
+    I suppose one'd want to study a bit the relationship between the two.
+    <fr:link
+href="millies-zeilberger-2015.xml"
+type="local"
+addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-269.xml b/_site/forest/unstable-269.xml
index 8ecaad81..681fae01 100644
--- a/_site/forest/unstable-269.xml
+++ b/_site/forest/unstable-269.xml
@@ -7,70 +7,28 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+root="false"><fr:frontmatter><fr:anchor>475</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title><fr:link
+href="hejohns-000J.xml"
+type="local"
+addr="hejohns-000J">Higher-order intuitionistic topos logic</fr:link> › syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-270.xml b/_site/forest/unstable-270.xml
index 1260d386..11cdeecd 100644
--- a/_site/forest/unstable-270.xml
+++ b/_site/forest/unstable-270.xml
@@ -7,48 +7,28 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>506</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title><fr:link
-href="dummett-91.xml"
+root="false"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title><fr:link
+href="hejohns-000J.xml"
 type="local"
-addr="dummett-91">The Logical Basis of Metaphysics</fr:link> › Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+addr="hejohns-000J">Higher-order intuitionistic topos logic</fr:link> › semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>504</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>468</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-271.xml b/_site/forest/unstable-271.xml
index e7453f8c..5727dd24 100644
--- a/_site/forest/unstable-271.xml
+++ b/_site/forest/unstable-271.xml
@@ -7,27 +7,18 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
-    </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
-    </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
+root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
     </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -35,34 +26,89 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
-    </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -70,26 +116,199 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
     </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-272.xml b/_site/forest/unstable-272.xml
index f83f9784..c1fb5a71 100644
--- a/_site/forest/unstable-272.xml
+++ b/_site/forest/unstable-272.xml
@@ -7,28 +7,52 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
+root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
-    </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
-    </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
     </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -36,34 +60,89 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>303</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
-    </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -71,26 +150,203 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
     </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-273.xml b/_site/forest/unstable-273.xml
index 6c63ffe1..ba35dd6d 100644
--- a/_site/forest/unstable-273.xml
+++ b/_site/forest/unstable-273.xml
@@ -7,13 +7,37 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
+root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
     </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -21,23 +45,286 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>305</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-274.xml b/_site/forest/unstable-274.xml
index 52b85734..386b2176 100644
--- a/_site/forest/unstable-274.xml
+++ b/_site/forest/unstable-274.xml
@@ -7,12 +7,28 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>311</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
+root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
     </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -20,19 +36,39 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -40,13 +76,246 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-275.xml b/_site/forest/unstable-275.xml
index ad785297..96686890 100644
--- a/_site/forest/unstable-275.xml
+++ b/_site/forest/unstable-275.xml
@@ -7,13 +7,21 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
+root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
     </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -21,19 +29,39 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
     </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -41,13 +69,246 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-276.xml b/_site/forest/unstable-276.xml
index 31255f8e..58df545f 100644
--- a/_site/forest/unstable-276.xml
+++ b/_site/forest/unstable-276.xml
@@ -7,288 +7,399 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="subb-000E.xml"
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
     </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
 type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+href="unstable-276.xml"
+type="local"
+addr="#276"><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
+show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-277.xml b/_site/forest/unstable-277.xml
index ff147a70..07ce2a69 100644
--- a/_site/forest/unstable-277.xml
+++ b/_site/forest/unstable-277.xml
@@ -7,377 +7,452 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title><fr:link
+href="unstable-276.xml"
+type="local"
+addr="#276"><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title><fr:link
+href="hejohns-000G.xml"
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="subb-000E.xml"
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
     </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
-type="local"
-addr="#277">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
+root="false"><fr:frontmatter><fr:anchor>400</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+href="unstable-276.xml"
+type="local"
+addr="#276"><fr:link
+href="hejohns-000G.xml"
+type="local"
+addr="hejohns-000G">What is an elementary topos?</fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="#277">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-278.xml b/_site/forest/unstable-278.xml
index 7f9586b3..8b54ad0f 100644
--- a/_site/forest/unstable-278.xml
+++ b/_site/forest/unstable-278.xml
@@ -7,146 +7,70 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
+root="false"><fr:frontmatter><fr:anchor>449</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title><fr:link
+href="hejohns-000C.xml"
 type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
+addr="hejohns-000C">Elementary² toposes</fr:link> › elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
-            <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
-                </fr:li>
-                <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>447</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
@@ -156,323 +80,132 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
-            <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
-                </fr:li>
-                <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>585</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
-type="local"
-addr="#277">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>587</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
-href="unstable-277.xml"
-type="local"
-addr="#277">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#277</fr:parent></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>589</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#267</fr:addr><fr:route>unstable-267.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
 </fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
@@ -480,592 +213,325 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>354</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
-type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
 type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
-</fr:p><fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
-href="hejohns-0007.xml"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
 type="local"
-addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
-type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
-type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
-type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
-</fr:p><fr:tree
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-279.xml b/_site/forest/unstable-279.xml
new file mode 100644
index 00000000..205a84fb
--- /dev/null
+++ b/_site/forest/unstable-279.xml
@@ -0,0 +1,1052 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title><fr:link
+href="hejohns-000C.xml"
+type="local"
+addr="hejohns-000C">Elementary² toposes</fr:link> › elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary² toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>#278</fr:addr><fr:route>unstable-278.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#279</fr:addr><fr:route>unstable-279.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+href="marquis-reyes-2012.xml"
+type="local"
+addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+href="lawvere-1970.xml"
+type="local"
+addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex>Set</fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+href="mclarty-1990b.xml"
+type="local"
+addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#271</fr:addr><fr:route>unstable-271.xml</fr:route><fr:title>Topos à la <fr:link
+href="makkai-reyes-1977.xml"
+type="local"
+addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#272</fr:addr><fr:route>unstable-272.xml</fr:route><fr:title>Topos à la <fr:link
+href="goldblatt-2006.xml"
+type="local"
+addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+href="https://math.stackexchange.com/q/1845697"
+type="external">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#273</fr:addr><fr:route>unstable-273.xml</fr:route><fr:title>Topos à la <fr:link
+href="johnstone-2002.xml"
+type="local"
+addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#274</fr:addr><fr:route>unstable-274.xml</fr:route><fr:title>Topos à la <fr:link
+href="awodey-2010.xml"
+type="local"
+addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#275</fr:addr><fr:route>unstable-275.xml</fr:route><fr:title>Topos à la <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#276</fr:addr><fr:route>unstable-276.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#277</fr:addr><fr:route>unstable-277.xml</fr:route><fr:title>Topos à la <fr:link
+href="maclane-moerdijk-1994.xml"
+type="local"
+addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#276</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
+            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
+            isomorphisms, each natural in <fr:tex>A</fr:tex>
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
+            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
+            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
+            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
+            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+href="sub-000G.xml"
+type="local"
+addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
+            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
+    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+    <fr:embedded-tex
+hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+         \begin {tikzcd}
+            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
+            A  \ar [r, "χ_s"'] &amp; Ω
+         \end {tikzcd}
+    </fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex>true</fr:tex> is trivially monic since every global element is
+    trivially monic.
+    Recall that "monos are stable under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+href="blass-1988.xml"
+type="local"
+addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+href="mclarty-1992.xml"
+type="local"
+addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>#269</fr:addr><fr:route>unstable-269.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#270</fr:addr><fr:route>unstable-270.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#262</fr:addr><fr:route>unstable-262.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
+        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex>1</fr:tex> only.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+href="lawvere-1970.pdf"
+type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#260</fr:addr><fr:route>unstable-260.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="mclarty-1990b.pdf"
+type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+href="ronnie-chen.xml"
+type="local"
+addr="ronnie-chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+href="marquis-reyes-2012.pdf"
+type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>439</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-280.xml b/_site/forest/unstable-280.xml
new file mode 100644
index 00000000..4d79a4cf
--- /dev/null
+++ b/_site/forest/unstable-280.xml
@@ -0,0 +1,353 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>773</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>771</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
+href="sub-000D.xml"
+type="local"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    However, unlike <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>,
+    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-281.xml b/_site/forest/unstable-281.xml
new file mode 100644
index 00000000..ccfdf640
--- /dev/null
+++ b/_site/forest/unstable-281.xml
@@ -0,0 +1,96 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>777</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>775</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#281</fr:addr><fr:route>unstable-281.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
+        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
+        Then <fr:tex>f</fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
+        Then <fr:tex>u</fr:tex> monotone as well.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
+        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
+        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+href="sub-0006.xml"
+type="local"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+        Thus <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
+        The <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-282.xml b/_site/forest/unstable-282.xml
new file mode 100644
index 00000000..054186d5
--- /dev/null
+++ b/_site/forest/unstable-282.xml
@@ -0,0 +1,76 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>497</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>496</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#282</fr:addr><fr:route>unstable-282.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+href="friedman-1975.pdf"
+type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-283.xml b/_site/forest/unstable-283.xml
new file mode 100644
index 00000000..6f683b73
--- /dev/null
+++ b/_site/forest/unstable-283.xml
@@ -0,0 +1,54 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>715</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title><fr:link
+href="dummett-91.xml"
+type="local"
+addr="dummett-91">The Logical Basis of Metaphysics</fr:link> › Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>713</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>#283</fr:addr><fr:route>unstable-283.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+href="zeilberger"
+type="external">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-284.xml b/_site/forest/unstable-284.xml
new file mode 100644
index 00000000..c93f4e15
--- /dev/null
+++ b/_site/forest/unstable-284.xml
@@ -0,0 +1,95 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>499</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>498</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-285.xml b/_site/forest/unstable-285.xml
new file mode 100644
index 00000000..470746f2
--- /dev/null
+++ b/_site/forest/unstable-285.xml
@@ -0,0 +1,96 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>501</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>500</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#284</fr:addr><fr:route>unstable-284.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#285</fr:addr><fr:route>unstable-285.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-286.xml b/_site/forest/unstable-286.xml
new file mode 100644
index 00000000..f22c1e1f
--- /dev/null
+++ b/_site/forest/unstable-286.xml
@@ -0,0 +1,43 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>504</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>503</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#286</fr:addr><fr:route>unstable-286.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+href="blass-2003.pdf"
+type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-287.xml b/_site/forest/unstable-287.xml
new file mode 100644
index 00000000..61787aa3
--- /dev/null
+++ b/_site/forest/unstable-287.xml
@@ -0,0 +1,52 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>506</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>505</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-288.xml b/_site/forest/unstable-288.xml
new file mode 100644
index 00000000..74e52157
--- /dev/null
+++ b/_site/forest/unstable-288.xml
@@ -0,0 +1,53 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>508</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>507</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#287</fr:addr><fr:route>unstable-287.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:taxon>Notes</fr:taxon><fr:addr>#288</fr:addr><fr:route>unstable-288.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
+href="blass-1988.pdf"
+type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-289.xml b/_site/forest/unstable-289.xml
new file mode 100644
index 00000000..930bda4b
--- /dev/null
+++ b/_site/forest/unstable-289.xml
@@ -0,0 +1,294 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>782</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>780</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>779</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-290.xml b/_site/forest/unstable-290.xml
new file mode 100644
index 00000000..1843f6a3
--- /dev/null
+++ b/_site/forest/unstable-290.xml
@@ -0,0 +1,383 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>791</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>789</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>785</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
+type="local"
+addr="#290">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>787</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
+type="local"
+addr="#290">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>784</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-291.xml b/_site/forest/unstable-291.xml
new file mode 100644
index 00000000..24e3e919
--- /dev/null
+++ b/_site/forest/unstable-291.xml
@@ -0,0 +1,1071 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<?xml-stylesheet type="text/xsl" href="default.xsl"?>
+<fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>806</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>804</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>#289</fr:addr><fr:route>unstable-289.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#290</fr:addr><fr:route>unstable-290.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+        <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">original theorem</fr:link> is stated in terms of
+        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>#291</fr:addr><fr:route>unstable-291.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>
+        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Then <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
+        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+href="subbb-000E.xml"
+type="local"
+addr="subbb-000E">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
+                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
+                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
+                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
+                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+href="hejohns-000D.xml"
+type="local"
+addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
+                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+href="subb-000E.xml"
+type="local"
+addr="subb-000E">I</fr:link> expresses that
+                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
+                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    <fr:embedded-tex
+hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                         \begin {tikzcd}
+                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
+                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
+                         \end {tikzcd}
+                    </fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
+                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the
+                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
+                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+href="subb-0006.xml"
+type="local"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>794</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
+type="local"
+addr="#290">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
+                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>796</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
+href="unstable-290.xml"
+type="local"
+addr="#290">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#290</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+             \begin {tikzcd}
+                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
+                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
+             \end {tikzcd}
+        </fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>798</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+href="street-2012.xml"
+type="local"
+addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:addr>#280</fr:addr><fr:route>unstable-280.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
+            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
+            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
+            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
+            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
+    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
+href="sub-000D.xml"
+type="local"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+    However, unlike <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>,
+    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>800</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+href="hejohns-0007.xml"
+type="local"
+addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
+            Then naturality of <fr:tex>π</fr:tex> requires that
+            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
+                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>802</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+href="hejohns-0006.xml"
+type="local"
+addr="hejohns-0006">the thin case</fr:link>.
+    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
+            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
+            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
+            This is our candidate for the <fr:tex>F</fr:tex>-action.
+            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
+            square in <fr:tex>Set</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
+                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
+            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+href="subb-000D.xml"
+type="local"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
+            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
+            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
+            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
+            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}
+                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
+                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
+            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
+            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
+href="sub-000D.xml"
+type="local"
+addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
+            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+href="subbb-000D.xml"
+type="local"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
+            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
+            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
+            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                 \begin {tikzcd}
+                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
+                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                 \end {tikzcd}
+            </fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
+            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
+            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
+            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
+            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            <fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
+                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
+                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
+                     \end {tikzcd}
+                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
+            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
+            that <fr:tex>F' =  \overline {F}</fr:tex>.
+            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
+            of <fr:tex>π</fr:tex> is exactly <fr:link
+href="subbbb-000D"
+type="external">this square</fr:link>, and forces
+            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
+type="external">The nlab page is here</fr:link>.
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+href="hejohns.xml"
+type="local"
+addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
+        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+href="sub-0007.xml"
+type="local"
+addr="sub-0007">the short proof</fr:link>.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>793</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:addr>#253</fr:addr><fr:route>unstable-253.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/van-heijenoort-1967.xml b/_site/forest/van-heijenoort-1967.xml
index 2a0e4f89..7c7873c5 100644
--- a/_site/forest/van-heijenoort-1967.xml
+++ b/_site/forest/van-heijenoort-1967.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>696</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -16,7 +16,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>486</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>695</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 href="quine-1960.xml"
diff --git a/_site/forest/von-neumann-1925.xml b/_site/forest/von-neumann-1925.xml
index 7e2c3904..0cf13203 100644
--- a/_site/forest/von-neumann-1925.xml
+++ b/_site/forest/von-neumann-1925.xml
@@ -7,7 +7,7 @@ numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>694</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 href="quine-1960.xml"
@@ -25,7 +25,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>690</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -52,7 +52,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>483</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>692</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -69,7 +69,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
@@ -135,7 +135,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>478</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"><fr:frontmatter><fr:anchor>687</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter></fr:tree><fr:tree
 xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
@@ -144,7 +144,7 @@ numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>479</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
+root="false"><fr:frontmatter><fr:anchor>688</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:contributor><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
@@ -161,7 +161,7 @@ numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"><fr:frontmatter><fr:anchor>591</fr:anchor><fr:taxon>Email</fr:taxon><fr:addr>#257</fr:addr><fr:route>unstable-257.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
 type="local"
 addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
diff --git a/_site/index.html b/_site/index.html
index 9c82f5c1..8d407d37 100644
--- a/_site/index.html
+++ b/_site/index.html
@@ -129,7 +129,7 @@ <h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/notes/1-25.html b/_site/notes/1-25.html
index 23c8435c..9aa7b32f 100644
--- a/_site/notes/1-25.html
+++ b/_site/notes/1-25.html
@@ -307,7 +307,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/notes/2-08.html b/_site/notes/2-08.html
index ef7282f7..932088a6 100644
--- a/_site/notes/2-08.html
+++ b/_site/notes/2-08.html
@@ -291,7 +291,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/posts.html b/_site/posts.html
index 360b4d7c..10379553 100644
--- a/_site/posts.html
+++ b/_site/posts.html
@@ -432,7 +432,7 @@ <h3>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/reviews.html b/_site/reviews.html
index dbf3b10d..ee308f47 100644
--- a/_site/reviews.html
+++ b/_site/reviews.html
@@ -69,8 +69,8 @@ <h1 class="post-title">reviews</h1>
 
 <p>Shoutout to kaobook.</p>
 
-<style> .pdf-embed-wrap-5e4d226d-81f4-4aa5-84d8-864e25897309 { display: flex; flex-direction: column; width: 100%; height: 650px; } .pdf-embed-container-5e4d226d-81f4-4aa5-84d8-864e25897309 { height: 100%; } .pdf-link-5e4d226d-81f4-4aa5-84d8-864e25897309 { background-color: white; text-align: center; border-style: solid; } .pdf-embed-container-5e4d226d-81f4-4aa5-84d8-864e25897309 iframe { width: 100%; height: 100%; } </style>
-<div class="pdf-embed-wrap-5e4d226d-81f4-4aa5-84d8-864e25897309"> <div class="pdf-link-5e4d226d-81f4-4aa5-84d8-864e25897309"> <a href="/files/sample.pdf" target="_blank">View PDF</a> </div> <div class="pdf-embed-container-5e4d226d-81f4-4aa5-84d8-864e25897309"> <iframe src="/files/sample.pdf" frameborder="0" allowfullscreen=""></iframe> </div> </div>
+<style> .pdf-embed-wrap-c3725f54-3f36-486d-a3da-8d1148ebe472 { display: flex; flex-direction: column; width: 100%; height: 650px; } .pdf-embed-container-c3725f54-3f36-486d-a3da-8d1148ebe472 { height: 100%; } .pdf-link-c3725f54-3f36-486d-a3da-8d1148ebe472 { background-color: white; text-align: center; border-style: solid; } .pdf-embed-container-c3725f54-3f36-486d-a3da-8d1148ebe472 iframe { width: 100%; height: 100%; } </style>
+<div class="pdf-embed-wrap-c3725f54-3f36-486d-a3da-8d1148ebe472"> <div class="pdf-link-c3725f54-3f36-486d-a3da-8d1148ebe472"> <a href="/files/sample.pdf" target="_blank">View PDF</a> </div> <div class="pdf-embed-container-c3725f54-3f36-486d-a3da-8d1148ebe472"> <iframe src="/files/sample.pdf" frameborder="0" allowfullscreen=""></iframe> </div> </div>
 
   </div>
 
@@ -99,7 +99,7 @@ <h1 class="post-title">reviews</h1>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction.html b/_site/site-construction.html
index 9e6a0c35..2972df7e 100644
--- a/_site/site-construction.html
+++ b/_site/site-construction.html
@@ -124,7 +124,7 @@ <h3 id="specs">specs</h3>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/05/katex-test.html b/_site/site-construction/2023/04/05/katex-test.html
index 09885e8b..bc4791d0 100644
--- a/_site/site-construction/2023/04/05/katex-test.html
+++ b/_site/site-construction/2023/04/05/katex-test.html
@@ -140,7 +140,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/05/last-modified-time-test.html b/_site/site-construction/2023/04/05/last-modified-time-test.html
index 22bb0b34..152d643c 100644
--- a/_site/site-construction/2023/04/05/last-modified-time-test.html
+++ b/_site/site-construction/2023/04/05/last-modified-time-test.html
@@ -128,7 +128,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/05/page-sorting.html b/_site/site-construction/2023/04/05/page-sorting.html
index 46f2253f..2907e796 100644
--- a/_site/site-construction/2023/04/05/page-sorting.html
+++ b/_site/site-construction/2023/04/05/page-sorting.html
@@ -125,7 +125,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/05/welcome-to-jekyll.html b/_site/site-construction/2023/04/05/welcome-to-jekyll.html
index 729076eb..6a8c3865 100644
--- a/_site/site-construction/2023/04/05/welcome-to-jekyll.html
+++ b/_site/site-construction/2023/04/05/welcome-to-jekyll.html
@@ -122,7 +122,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/07/underscore-plugins.html b/_site/site-construction/2023/04/07/underscore-plugins.html
index f046b65a..dcedc90e 100644
--- a/_site/site-construction/2023/04/07/underscore-plugins.html
+++ b/_site/site-construction/2023/04/07/underscore-plugins.html
@@ -138,7 +138,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/05/14/darcs.html b/_site/site-construction/2023/05/14/darcs.html
index a71bc023..a3a1bf7d 100644
--- a/_site/site-construction/2023/05/14/darcs.html
+++ b/_site/site-construction/2023/05/14/darcs.html
@@ -117,7 +117,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/11/06/live-reload.html b/_site/site-construction/2023/11/06/live-reload.html
index 95cd1ab5..3ef8d442 100644
--- a/_site/site-construction/2023/11/06/live-reload.html
+++ b/_site/site-construction/2023/11/06/live-reload.html
@@ -118,7 +118,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-01</p>
+        <p>site version: 2024-06-03</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/sitemap.xml b/_site/sitemap.xml
index 7dc3950f..5e7559af 100644
--- a/_site/sitemap.xml
+++ b/_site/sitemap.xml
@@ -852,55 +852,71 @@
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/blass-1988.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/blass-2003.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/carette-kiselyov-shan-2007.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/friedman-1975.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/hoffman-1999.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/lambek-1989.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
+</url>
+<url>
+<loc>https://hejohns.github.io/forest/lawvere-1970.pdf</loc>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/maddy-2019.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
+</url>
+<url>
+<loc>https://hejohns.github.io/forest/marquis-reyes-2012.pdf</loc>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
+</url>
+<url>
+<loc>https://hejohns.github.io/forest/mclarty-1990a.pdf</loc>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
+</url>
+<url>
+<loc>https://hejohns.github.io/forest/mclarty-1990b.pdf</loc>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/millies-zeilberger-2015.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:50-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/pfenning-elliott-1988.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:51-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/quine-1960.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:51-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/statman-1985.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:51-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/sterling-2022.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:51-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/wadler-1989-ocr.pdf</loc>
-<lastmod>2024-06-01T16:41:42-04:00</lastmod>
+<lastmod>2024-06-03T05:14:51-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/google1be2851b772a6a30.html</loc>