From 70403280cecb99d809ce0cd446db45f56bc77cfb Mon Sep 17 00:00:00 2001 From: Johnson He Date: Mon, 3 Jun 2024 05:15:33 -0400 Subject: [PATCH] bump --- _forest | 2 +- _site/2023/04/09/gsi-applications.html | 6 +- _site/2023/04/10/miserliness.html | 2 +- _site/2023/04/14/rasterize-pdfs.html | 2 +- _site/2023/04/23/scripting.html | 2 +- _site/2023/05/02/chili-oil.html | 2 +- _site/2023/05/07/nonstandard-analysis.html | 2 +- _site/2023/05/10/linux-graphics-modules.html | 2 +- _site/2023/07/24/book-catalogue.html | 2 +- _site/2023/08/03/nic-bonding.html | 2 +- _site/2023/10/10/tortoise-tts.html | 2 +- _site/2023/10/29/coordinating-processes.html | 2 +- _site/2023/11/07/ble-sh.html | 2 +- _site/2023/12/19/icecc.html | 2 +- _site/2023/12/20/xmonad.html | 2 +- _site/2024/01/08/thunderbird-and-nix.html | 2 +- _site/2024/01/16/mellanox.html | 2 +- _site/2024/02/08/natural-models.html | 2 +- _site/2024/02/12/noctua.html | 2 +- _site/2024/02/24/vim.html | 2 +- _site/2024/03/10/deoplete.html | 2 +- _site/2024/04/11/fountain-pen-nibs.html | 2 +- _site/404.html | 2 +- .../2023/05/18/equivalence-of-categories.html | 2 +- _site/category/category-theory/index.html | 2 +- _site/category/site-construction/index.html | 2 +- _site/feed.xml | 2 +- _site/forest.html | 2 +- _site/forest/awodey-2010.xml | 10 + _site/forest/blass-1988.xml | 120 +- _site/forest/blass-2003.xml | 4 +- _site/forest/carette-kiselyov-shan-2007.xml | 12 +- _site/forest/dummett-91.xml | 4 +- _site/forest/forest.json | 2 +- _site/forest/friedman-1975.xml | 10 +- _site/forest/goldblatt-2006.xml | 17 + _site/forest/hejohns-0001.xml | 18 +- _site/forest/hejohns-0002.xml | 12 +- _site/forest/hejohns-0003.xml | 6 +- _site/forest/hejohns-0004.xml | 10 +- _site/forest/hejohns-0005.xml | 10 +- _site/forest/hejohns-0006.xml | 92 +- _site/forest/hejohns-0007.xml | 98 +- _site/forest/hejohns-0008.xml | 52 +- _site/forest/hejohns-0009.xml | 8 +- _site/forest/hejohns-000A.xml | 2 +- _site/forest/hejohns-000B.xml | 4 +- 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insertions(+), 12939 deletions(-) create mode 100644 _site/forest/awodey-2010.xml create mode 100644 _site/forest/goldblatt-2006.xml create mode 100644 _site/forest/hejohns-000F.xml create mode 100644 _site/forest/hejohns-000G.xml create mode 100644 _site/forest/hejohns-000H.xml create mode 100644 _site/forest/hejohns-000I.xml create mode 100644 _site/forest/hejohns-000J.xml create mode 100644 _site/forest/johnstone-2002.xml create mode 100644 _site/forest/lawvere-1970.pdf create mode 100644 _site/forest/lawvere-1970.xml create mode 100644 _site/forest/maclane-moerdijk-1994.xml create mode 100644 _site/forest/makkai-reyes-1977.xml create mode 100644 _site/forest/marquis-reyes-2012.pdf create mode 100644 _site/forest/marquis-reyes-2012.xml create mode 100644 _site/forest/mclarty-1990a.pdf create mode 100644 _site/forest/mclarty-1990a.xml create mode 100644 _site/forest/mclarty-1990b.pdf create mode 100644 _site/forest/mclarty-1990b.xml create mode 100644 _site/forest/resources/e02927923b00b58796778a2184dd7584.svg create mode 100644 _site/forest/ronnie-chen.xml create mode 100644 _site/forest/sub-000G.xml create mode 100644 _site/forest/unstable-279.xml create mode 100644 _site/forest/unstable-280.xml create mode 100644 _site/forest/unstable-281.xml create mode 100644 _site/forest/unstable-282.xml create mode 100644 _site/forest/unstable-283.xml create mode 100644 _site/forest/unstable-284.xml create mode 100644 _site/forest/unstable-285.xml create mode 100644 _site/forest/unstable-286.xml create mode 100644 _site/forest/unstable-287.xml create mode 100644 _site/forest/unstable-288.xml create mode 100644 _site/forest/unstable-289.xml create mode 100644 _site/forest/unstable-290.xml create mode 100644 _site/forest/unstable-291.xml 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 @@

ENGR 101 (F23)

- -
View PDF
+ +
View PDF

EECS 376 (F23)

outcome: rejected. No response.

too long version

@@ -153,7 +153,7 @@

related

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    a pure waste of space

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    a pure waste of space

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    a pure waste of space

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    a pure waste of space

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    a pure waste of space

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    related

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    a pure waste of space

    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 @@

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    a pure waste of space

    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 @@

    Archive of posts with category 'category-theory'

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    a pure waste of space

    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 @@

    Archive of posts with category 'site-construction'

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    a pure waste of space

    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 @@ -Jekyll2024-06-01T16:41:42-04:00https://hejohns.github.io/feed.xmlhejohns’ jekyll sitea pure waste of spaceJohnson Hehejohns α umich δ edufountain pen nibs2024-04-11T00:00:00-04:002024-04-11T17:48:22-04:00https://hejohns.github.io/2024/04/11/fountain-pen-nibsSeveral weeks ago, I lost my other TWSBI 580 (technically, the 580 ALR) +Jekyll2024-06-03T05:14:53-04:00https://hejohns.github.io/feed.xmlhejohns’ jekyll sitea pure waste of spaceJohnson Hehejohns α umich δ edufountain pen nibs2024-04-11T00:00:00-04:002024-04-11T17:48:22-04:00https://hejohns.github.io/2024/04/11/fountain-pen-nibsSeveral weeks ago, I lost my other TWSBI 580 (technically, the 580 ALR) which meant that I didn’t have any EF nibs left1.

    Through the process of sourcing nibs, I learned a lot about the details so:

    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 @@

  • (the answer is yes, you should.)
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    a pure waste of space

    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 @@ + + +339Referenceawodey-2010awodey-2010.xmlCategory Theory202462202463 \ 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">463Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461676Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461308Abstract#274unstable-274.xml202461202461blass-1988 +root="false">299Abstract#287unstable-287.xml202461202461blass-1988 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">309Notes#275unstable-275.xml202461202461202461202461blass-1988 +root="false">300Notes#288unstable-288.xml202461202461202461202461blass-1988 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">461Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202461672hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He - The following are some basic (alt. elementary) questions, whose "elevator - pitch" answers ought to roll off the tongue: - - What is a(n elementary) topos? - - - 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? - - - What is this "Kripke-Joyal semantics"? - Is this the same thing as the so-called "forcing semantics"? - - - What is this "Mitchell-Bénabou language"? - Is it the same thing as "topos logic"? - - Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. - - Some more difficult questions we do not attempt to answer at the moment: - - What is a Grothendieck topos, and what's the difference v. - elementary toposes? - - - What is a "geometric logic" and a "geometric morphism"? - And what's the difference v. a "logical morphism"? - - - In what sense is "Kripke-Joyal semantics" a "forcing semantics"? - (What is forcing??) - - - And everything else… - - Many object to the name "higher-order intuitionistic logic". - Topoi and Computation says "Strictly speaking, the phrase 'higher-order +addr="blass-1988">Topoi and Computation 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." - maclarty-1992 says "TODO". + underlying logic." (p. 2). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +674hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. \ 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">456Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461667Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461305Abstract#273unstable-273.xml202461202461blass-2003 +root="false">502Abstract#286unstable-286.xml202461202461blass-2003 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">435Referencecarette-kiselyov-shan-2007carette-kiselyov-shan-2007.xmlFinally Tagless, Partially Evaluated2024524202461646Referencecarette-kiselyov-shan-2007carette-kiselyov-shan-2007.xmlFinally Tagless, Partially Evaluated2024524202461283Abstract#271unstable-271.xml2024524202461carette-kiselyov-shan-2007 +root="false">480Abstract#284unstable-284.xml2024524202461carette-kiselyov-shan-2007 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">284Notes#272unstable-272.xml20245242024612024525202461carette-kiselyov-shan-2007 +root="false">481Notes#285unstable-285.xml20245242024612024525202461carette-kiselyov-shan-2007 This is the APLAS version (conference paper). There is a roughly twice as long journal version. @@ -71,14 +71,14 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">434Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461645Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461279Abstract#254unstable-254.xml2024524202461sterling-2022 +root="false">476Abstract#254unstable-254.xml2024524202461sterling-2022 Logical relations are the main tool for proving positive 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">280#255unstable-255.xml20245242024612024525202461sterling-2022 +root="false">477#255unstable-255.xml20245242024612024525202461sterling-2022 I can't quite follow the phase distinction and only have a very vague idea of what's going on starting at p. 9. 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">508Referencedummett-91dummett-91.xmlThe Logical Basis of Metaphysics20245122024513717Referencedummett-91dummett-91.xmlThe Logical Basis of Metaphysics20245122024513Johnson He395#270unstable-270.xmlStability20245122024513599#283unstable-283.xmlStability20245122024513Johnson Hedummett-91 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, 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+++ b/_site/forest/friedman-1975.xml @@ -7,14 +7,14 @@ numbered="true" show-heading="true" show-metadata="true" expanded="true" -root="false">428Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461639Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461288Abstract#269unstable-269.xml2024513202461friedman-1975 +root="false">485Abstract#282unstable-282.xml2024513202461friedman-1975 The λ-calculus, in both its typed and untyped forms, has primarily been regarded as an attempt to formalize the concept of rule or process, 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">426Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461637Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461https://arxiv.org/pdf/1610.01213. The nice introduction informed me of 427Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461638Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461290Abstract#256unstable-256.xml2024513202461statman-1985 +root="false">487Abstract#256unstable-256.xml2024513202461statman-1985 In this note we shall try to place Friedman's remarkable little paper ( + +357Referencegoldblatt-2006goldblatt-2006.xmlTopoi: The Categorial Analysis of Logic202462202463 + "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." + \ 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">763Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456998Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He @@ -20,7 +20,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="false" -root="false">322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He @@ -40,7 +40,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He @@ -55,7 +55,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He @@ -68,7 +68,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">761Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461996Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461Johnson HeThis is a fundamental notion. @@ -86,7 +86,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">326Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456521Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He @@ -99,7 +99,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="false" -root="false">322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He @@ -119,7 +119,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He @@ -134,7 +134,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He 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">755Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456990Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He @@ -27,7 +27,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He @@ -40,7 +40,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">753Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456988Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He @@ -53,7 +53,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="false" -root="false">322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He @@ -73,7 +73,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He @@ -88,7 +88,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He 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">751Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456986Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He @@ -20,7 +20,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">749Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456984Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He @@ -40,7 +40,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He 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">759Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456994Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He @@ -20,7 +20,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">757Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456992Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He @@ -33,7 +33,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="false" -root="false">322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He @@ -53,7 +53,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He @@ -68,7 +68,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He 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">472Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461681Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461Johnson HeThis is a fundamental notion. @@ -25,7 +25,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">326Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456521Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He @@ -38,7 +38,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="false" -root="false">322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He @@ -58,7 +58,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He @@ -73,7 +73,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He 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">648Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511857Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -22,7 +22,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -39,7 +39,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -58,7 +58,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -83,7 +83,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">646Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510855Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510Johnson He336Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511531Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -108,7 +108,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -125,7 +125,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -144,7 +144,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -169,7 +169,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">356Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511551Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -184,7 +184,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -212,7 +212,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -226,7 +226,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -305,7 +305,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -499,7 +499,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">370Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511565Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E364Proposition#277unstable-277.xml20245112024511559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -554,10 +554,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -572,10 +572,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -590,7 +590,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">366Proof#278unstable-278.xml20245112024511561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -729,7 +729,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E @@ -748,7 +748,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">640Proof corollarysubb-0006subb-0006.xml2024572024511849Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -773,7 +773,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">642Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511851Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -788,7 +788,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -816,7 +816,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -830,7 +830,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -909,7 +909,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -1103,7 +1103,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">644Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511853Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -1141,7 +1141,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -1328,7 +1328,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">639Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 +root="false">848Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 The first point: 391Abstract#253unstable-253.xml2024572024522street-2012 +root="false">595Abstract#253unstable-253.xml2024572024522street-2012 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">622Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511831Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -22,7 +22,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -50,7 +50,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -64,7 +64,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -143,7 +143,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -337,7 +337,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">620Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510829Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510Johnson He336Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511531Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -362,7 +362,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -379,7 +379,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -398,7 +398,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -423,7 +423,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">356Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511551Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -438,7 +438,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -466,7 +466,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -480,7 +480,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -559,7 +559,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -753,7 +753,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">370Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511565Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E364Proposition#277unstable-277.xml20245112024511559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -808,10 +808,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -826,10 +826,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -844,7 +844,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">366Proof#278unstable-278.xml20245112024511561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -983,7 +983,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E @@ -1002,7 +1002,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">618Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511827Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -1017,7 +1017,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -1034,7 +1034,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -1053,7 +1053,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -1078,7 +1078,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">616Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511825Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E364Proposition#277unstable-277.xml20245112024511559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -1133,10 +1133,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -1151,10 +1151,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -1169,7 +1169,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">366Proof#278unstable-278.xml20245112024511561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -1308,7 +1308,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E @@ -1327,7 +1327,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">615Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 +root="false">824Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 The first point: 391Abstract#253unstable-253.xml2024572024522street-2012 +root="false">595Abstract#253unstable-253.xml2024572024522street-2012 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">673Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510882Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510Johnson He336Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511531Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -32,7 +32,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -49,7 +49,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -68,7 +68,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -93,7 +93,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">356Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511551Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -108,7 +108,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -136,7 +136,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -150,7 +150,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -229,7 +229,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -423,7 +423,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">370Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511565Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E364Proposition#277unstable-277.xml20245112024511559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -478,10 +478,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -496,10 +496,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -514,7 +514,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">366Proof#278unstable-278.xml20245112024511561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -653,7 +653,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E @@ -672,7 +672,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">672Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 +root="false">881Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 The first point: 391Abstract#253unstable-253.xml2024572024522street-2012 +root="false">595Abstract#253unstable-253.xml2024572024522street-2012 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">470hejohns-0009hejohns-0009.xmlFibrations v. functors202458202461679hejohns-0009hejohns-0009.xmlFibrations v. functors202458202461Johnson He @@ -31,7 +31,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">468Referencemillies-zeilberger-2015millies-zeilberger-2015.xmlFunctors are Type Refinement Systems2024511202461677Referencemillies-zeilberger-2015millies-zeilberger-2015.xmlFunctors are Type Refinement Systems2024511202461Johnson He293Abstract#258unstable-258.xml2024511202461millies-zeilberger-2015 +root="false">490Abstract#258unstable-258.xml2024511202461millies-zeilberger-2015 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">294#259unstable-259.xmlSections 1-22024511202461491#259unstable-259.xmlSections 1-22024511202461Johnson Hemillies-zeilberger-2015What 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">432hejohns-000Ahejohns-000A.xmlFiberwise Structure202458202461643hejohns-000Ahejohns-000A.xmlFiberwise Structure202458202461Johnson He 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">446hejohns-000Bhejohns-000B.xmlLogical Frameworks as a meta-language2024527202461657hejohns-000Bhejohns-000B.xmlLogical Frameworks as a meta-language2024527202461Johnson He @@ -34,7 +34,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">445Referencevon-neumann-1925von-neumann-1925.xmlAn Axiomatization of Set Theory20245282024529 +root="false">656Referencevon-neumann-1925von-neumann-1925.xmlAn Axiomatization of Set Theory20245282024529 One of the early NBG papers. Maybe "the" NBG paper. My notes on this are currently in 466Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202461461Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He - The following are some basic (alt. elementary) questions, whose "elevator - pitch" answers ought to roll off the tongue: - - What is a(n elementary) topos? - - - 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? - - - What is this "Kripke-Joyal semantics"? - Is this the same thing as the so-called "forcing semantics"? - - - What is this "Mitchell-Bénabou language"? - Is it the same thing as "topos logic"? - - Following Johnson He330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) - Some more difficult questions we do not attempt to answer at the moment: - - What is a Grothendieck topos, and what's the difference v. - elementary toposes? - - - What is a "geometric logic" and a "geometric morphism"? - And what's the difference v. a "logical morphism"? - - - In what sense is "Kripke-Joyal semantics" a "forcing semantics"? - (What is forcing??) - - - And everything else… - - Many object to the name "higher-order intuitionistic logic". - The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation says "Strictly speaking, the phrase 'higher-order +addr="blass-1988">Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J451Referenceawodey-2010awodey-2010.xmlCategory Theory202462202463452Referencemakkai-reyes-1977makkai-reyes-1977.xmlFirst Order Categorical Logic202462202463 + I've heard/been told this is the book for, well, the subject in the + title, and Makkai completeness. +453Referencelawvere-1970lawvere-1970.xmlQuantifiers and Sheaves202461202463295Abstract#262unstable-262.xml202461202463lawvere-1970 + 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 S 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 external 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 S (BVM/S) 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". + + 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 (∈) left behind by the + process of accumulating (∪) the power set (\mathcal {P}) 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 1 only. + + I have not read this, and have no idea what "contradictions" he's referring to. maclarty-1992 says "TODO". -lawvere-1970.pdf454Referencemaclane-moerdijk-1994maclane-moerdijk-1994.xmlSheaves in Geometry and Logic: A First Introduction to Topos Theory202462202463455Referencejohnstone-2002johnstone-2002.xmlSketches of an Elephant: a topos theory compendium202462202463456Referencemclarty-1990bmclarty-1990b.xmlThe Uses and Abuses of the History of Topos Theory202461202463292Abstract#260unstable-260.xml202461202463mclarty-1990b + 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. + mclarty-1990b.pdf457Referencemarquis-reyes-2012marquis-reyes-2012.xmlThe history of categorical logic: 1963-1977202461202463 + I have not read this. This might be a good source, aside from Ronnie Chen, + where the viewpoint of "generalized algebraic semantics" is developed. +marquis-reyes-2012.pdf458Referencegoldblatt-2006goldblatt-2006.xmlTopoi: The Categorial Analysis of Logic202462202463 + "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." +464Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461459Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461465Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461460Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461308Abstract#274unstable-274.xml202461202461blass-1988 +root="false">299Abstract#287unstable-287.xml202461202461blass-1988 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">309Notes#275unstable-275.xml202461202461202461202461blass-1988 +root="false">300Notes#288unstable-288.xml202461202461202461202461blass-1988 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">670Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511879Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -45,7 +45,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -232,7 +232,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">668Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511877Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -247,7 +247,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -275,7 +275,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -289,7 +289,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -368,7 +368,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -562,7 +562,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">660sub-000Dsub-000D.xml20245102024511869sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D662subb-000Dsubb-000D.xml20245102024511871subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D664Moralsubbb-000Dsubbb-000D.xml20245102024511873Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -618,7 +618,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">666Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511875Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -633,7 +633,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -650,7 +650,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -669,7 +669,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -694,7 +694,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">658Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511867Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E364Proposition#277unstable-277.xml20245112024511559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -749,10 +749,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -767,10 +767,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -785,7 +785,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">366Proof#278unstable-278.xml20245112024511561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -924,7 +924,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E 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">637Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511846Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E364Proposition#277unstable-277.xml20245112024511559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -62,10 +62,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -80,10 +80,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -98,7 +98,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">366Proof#278unstable-278.xml20245112024511561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -237,7 +237,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E @@ -256,7 +256,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">635Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510844Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510Johnson He336Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511531Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -281,7 +281,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -298,7 +298,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -317,7 +317,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -342,7 +342,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">356Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511551Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -357,7 +357,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -385,7 +385,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -399,7 +399,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -478,7 +478,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -672,7 +672,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">370Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511565Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E364Proposition#277unstable-277.xml20245112024511559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -727,10 +727,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -745,10 +745,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -763,7 +763,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">366Proof#278unstable-278.xml20245112024511561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -902,7 +902,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E @@ -921,13 +921,13 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">625subb-000Esubb-000E.xml834subb-000Esubb-000E.xmlUntitled › I20245112024511Untitled › I20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -942,13 +942,13 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">627subbb-000Esubbb-000E.xml836subbb-000Esubbb-000E.xmlUntitled › II20245112024511Untitled › II20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -963,7 +963,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">629Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511838Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -978,7 +978,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -1006,7 +1006,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -1020,7 +1020,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -1099,7 +1099,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -1293,7 +1293,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">631Proofsub-0007sub-0007.xml840Proofsub-0007sub-0007.xmlPrecursor to The Core of Adjoint Functors › the short proof (sketch)2024572024511340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007633Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511842Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -1389,7 +1389,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -1576,7 +1576,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="true" -root="false">624Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 +root="false">833Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 The first point: 391Abstract#253unstable-253.xml2024572024522street-2012 +root="false">595Abstract#253unstable-253.xml2024572024522street-2012 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 @@ + + +336hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +328#279unstable-279.xmlElementary² toposes › elementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J334Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J298Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461301Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461299Abstract#287unstable-287.xml202461202461blass-1988 + 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. + 300Notes#288unstable-288.xml202461202461202461202461blass-1988 + 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. + blass-1988.pdf \ 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 @@ + + +421hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +417#279unstable-279.xmlElementary² toposes › elementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J419Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J415Definitionsub-000Gsub-000G.xmlWhat is an elementary topos? › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω408Referenceawodey-2010awodey-2010.xmlCategory Theory202462202463409Referencemakkai-reyes-1977makkai-reyes-1977.xmlFirst Order Categorical Logic202462202463 + I've heard/been told this is the book for, well, the subject in the + title, and Makkai completeness. +410Referencemaclane-moerdijk-1994maclane-moerdijk-1994.xmlSheaves in Geometry and Logic: A First Introduction to Topos Theory202462202463411Referencejohnstone-2002johnstone-2002.xmlSketches of an Elephant: a topos theory compendium202462202463412Referencemclarty-1990bmclarty-1990b.xmlThe Uses and Abuses of the History of Topos Theory202461202463292Abstract#260unstable-260.xml202461202463mclarty-1990b + 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. + mclarty-1990b.pdf413Referencegoldblatt-2006goldblatt-2006.xmlTopoi: The Categorial Analysis of Logic202462202463 + "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." +414Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461 \ 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 @@ + + +430hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +426#279unstable-279.xmlElementary² toposes › elementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J428Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J423Referencelawvere-1970lawvere-1970.xmlQuantifiers and Sheaves202461202463295Abstract#262unstable-262.xml202461202463lawvere-1970 + 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 S 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 external 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 S (BVM/S) 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". + + 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 (∈) left behind by the + process of accumulating (∪) the power set (\mathcal {P}) 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 1 only. + + I have not read this, and have no idea what "contradictions" he's referring to. + lawvere-1970.pdf424Referencemarquis-reyes-2012marquis-reyes-2012.xmlThe history of categorical logic: 1963-1977202461202463 + I have not read this. This might be a good source, aside from Ronnie Chen, + where the viewpoint of "generalized algebraic semantics" is developed. +marquis-reyes-2012.pdf425Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461299Abstract#287unstable-287.xml202461202461blass-1988 + 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. + 300Notes#288unstable-288.xml202461202461202461202461blass-1988 + 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. + blass-1988.pdf \ 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 @@ + + +473Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. +471hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +470Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461 \ 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 @@ + + +467hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J463#279unstable-279.xmlElementary² toposes › elementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J465Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J \ 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">746Personhejohnshejohns.xmlJohnson He202455202459My screen name is hejohns. This is my forest. +root="false">981Personhejohnshejohns.xmlJohnson He202455202459My screen name is hejohns. This is my forest. email: hejohns@umich.edu676hejohns-000Ahejohns-000A.xmlFiberwise Structure202458202461885hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He - Fibered Right Adjoints are automatically fibered? + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. 678hejohns-000Bhejohns-000B.xmlLogical Frameworks as a meta-language2024527202461887Definitionsub-000Gsub-000G.xmlWhat is an elementary topos? › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He - While reading Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω889Definition#271unstable-271.xmlAn Axiomatization of Set Theory, 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. - - That is, (classical) FoL is often the setting in which axiomatic - mathematical theories are formalized. Classic (no pun intended) examples - are ( \mathcal {L}_{group} = \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups})), ( \mathcal {L}_{DLO} = \{ ≤ \} , Th( \textsf {Dense Linear Orders})), ( \mathcal {L}_{sets} = \{ ∈ \} , ZFC). - (To be ultra-pedantic is a fruitless way, I've paired the language and theory.) - - "definite property" -What is an elementary topos? › Topos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 680#259unstable-259.xml891Definition#272unstable-272.xmlFunctors are Type Refinement Systems › Sections 1-22024511202461What is an elementary topos? › Topos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hemillies-zeilberger-2015What is a logical relation? - A LR is just a model of the syntax, used to prove ∀M statements. - Thus, it's usually just 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 just a clever choice of model, - to let us say something interesting/talk about what we care about. - - 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 two 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. - - This way, we can show that two different semantics and implementations - for a intset 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 Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." - 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? + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 893Definition#273unstable-273.xmlWhat is an elementary topos? › Topos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" - 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 topos is a properly cartesian closed category with a subobject + classifier." - 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." - Maybe I'm finally close to being able to understand what a Kripke - logical relation is. + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) 682Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202461895Definition#274unstable-274.xmlJohnson He - The following are some basic (alt. elementary) questions, whose "elevator - pitch" answers ought to roll off the tongue: - - What is a(n elementary) topos? - - - 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? - - - What is this "Kripke-Joyal semantics"? - Is this the same thing as the so-called "forcing semantics"? - - - What is this "Mitchell-Bénabou language"? - Is it the same thing as "topos logic"? - - Following What is an elementary topos? › Topos à la Elementary Categories, Elementary Toposes, we sketch elementary answers to these. - - Some more difficult questions we do not attempt to answer at the moment: - - What is a Grothendieck topos, and what's the difference v. - elementary toposes? - - - What is a "geometric logic" and a "geometric morphism"? - And what's the difference v. a "logical morphism"? - - - In what sense is "Kripke-Joyal semantics" a "forcing semantics"? - (What is forcing??) - - - And everything else… - - Many object to the name "higher-order intuitionistic logic". - Category Theory 8.17202461202463Topoi and Computation 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." - maclarty-1992 says "TODO". -Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 684hejohns-0009hejohns-0009.xmlFibrations v. functors202458202461897Definition#275unstable-275.xmlJohnson He - What is the actual logic when doing a logical relation? - Do you only actually need functors? Zeilberger's Functors as Type Refinement Systems, is a must read. - Then you lose the internal logic... maybe? Maybe you just have a different internal logic? - - 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. - What is an elementary topos? › Topos à la paperElementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 686Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461899#276unstable-276.xmlWhat is an elementary topos? › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson HeThis is a fundamental notion. - 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 notes on - realizability, - according to Loris it's relevant to Markov Decision Processes, and we - develop the view that it's a logician's view of fibrations. -Johnson Hehejohns-000G326Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He - Here is Blackburn et all's account: - - The concept first presented is that of Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + 901Definition#277unstable-277.xmlWhat is an elementary topos? › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + 903hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He - "The basic case": + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. - A relation Z (to not be confused with the accessibility relations R for - each Kripke frame) between two Kripke frames M and M' is a *bisimulation* iff - w Z w' ⟹ (w and w' are elementarily equivalent) - - - (forth) w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v' - - (backwards) w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson He - "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" - -. - A simulation is where we drop the backwards condition. -Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson He - Clearly the truth of ∧,∨,◇-formulas are preserved under simulations. - Actually it's shown that the class of formulas preserved under simulations are exactly these formulas. -Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 688Email#257unstable-257.xml20245282024529308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hequine-1960 - Thanks for sharing this nice little note, which sketches a combinatory - (classical) first-order logic. +addr="hejohns">Johnson Hehejohns-000G + "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:" - I was surprised by the mention of von Neumann 1925 in this snippet: + "A topos is a properly cartesian closed category with a subobject + classifier." - "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) - - 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. - - 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 class existence theorem. - - (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.) - - 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. - - 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. - - 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." - Johnson He + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) 690#270unstable-270.xml310Definition#274unstable-274.xmlTopos à la The Logical Basis of Metaphysics › Stability20245122024513Category Theory 8.17202461202463Johnson Hedummett-91 - The opening sentence is what zeilberger quotes on his work on - polarity. - - 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 Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." 692Referencedummett-91dummett-91.xmlThe Logical Basis of Metaphysics20245122024513312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson HeJohnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 395#270unstable-270.xmlStability20245122024513318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hedummett-91 - The opening sentence is what zeilberger quotes on his work on - polarity. - - 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. - Johnson Hehejohns-000G694Proofsub-0006sub-0006.xml2024572024511314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson Hehejohns-0006 - Suppose p₁ ≤ p₂. - Notice that p₂ ≤ u∘f(p₂), by the unit of the (thin) adjunction. - Then p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂) by the Hom bijection of the (thin) adjunction. - - (dual) - Suppose q₁ ≤ q₂. - Notice that f∘u(q₁) ≤ q₂ by the counit of the (thin) adjunction. - Then f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂) by the Hom bijection of the (thin) adjunction. - Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω696sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\ - X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - 316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +698Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E - (For V-enriched categories, ) If V is thin, all diagrams commute, - thus an adjunction core always determines an adjunction. - Set isn't thin, but 2 is, thus for posets, an adjunction core - gives rise to a thin adjunction, as remarked 320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from here. - I don't know how useful this lemma is for ordinary Homset categories. - Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +700Proof corollarysubb-0006subb-0006.xml2024572024511905hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson Hehejohns-0006 - Notice that the above proof does not use the information that u is monotone. - Thus Johnson Hethe above lemma holds where f and u are just a priori functions. - The Topoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If above lemma is in some sense the more - immediately natural question, but can be easily strengthened to - the form of this corollary. - Quantifiers and Sheaves 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. +702subb-000Dsubb-000D.xml20245102024511907#279unstable-279.xmlElementary² toposes › elementary answers2024531202463Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\ - X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"] - \end {tikzcd} - Johnson Hehejohns-000C704Moralsubbb-000Dsubbb-000D.xml20245102024511302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson Hehejohns-000D - Of course in hindsight, we could've (or even should've) - just defined \overline {F} to be - よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y) 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. - - This is the perspective taken in Johnson Hethe short proof. - Topoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +706subsub-0007subsub-0007.xml2024572024511321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +708subsubb-0007subsubb-0007.xml2024572024511304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(-), a) \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\ - X(-, U(a)) \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] & X(-, U(b)) - \end {tikzcd} - Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 710Proposition#267unstable-267.xml2024572024511306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-0007 - Suppose U : A → X is a functor, F : X .ob → A .ob a function, - and (πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X} a family of natural - isomorphisms. Then there is a unique extension \overline {F} of F such - that (πₓ)_{x ∈ X} is natural in x, witnessing \overline {F} ⊣ U. - - - Or dually, suppose that F is a functor, but U is just a - function on objects, and (πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}. - Then U uniquely extends to \overline {U}. - Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 712Proposition#268unstable-268.xml2024572024511308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-0006 - Suppose P and Q are posets, f : P → Q is a priori just a function, and u : P ← Q monotone, - such that f(x) ≤ a ⟺ x ≤ u(a). - Then f monotone as well. +addr="hejohns">Johnson Hehejohns-000G + "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:" - Or dually, suppose that f monotone, but u is a priori just a function. - Then u monotone as well. + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) 714Definition#276unstable-276.xml20245112024511310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000E"An adjunction core consists of" +addr="hejohns">Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that - functions U : A .ob → X .ob and F : X .ob → A .ob + 1. \mathcal {E} has all finite limits, + - a family of isomorphisms (π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}\mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 716Proposition#277unstable-277.xml20245112024511Johnson Hehejohns-000E - An adjunction core extends to an adjuction iff ∀a,b ∈ A. ∀x ∈ X. - 312Definition#275unstable-275.xmlTopos à la I commutes, and iff ∀x,y ∈ X. a ∈ A. Elementary Categories, Elementary Toposes 13.1202461202463II - commutes. - The adjunction is unique when it exists. +addr="hejohns">Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." - The original theorem is stated in terms of - \mathcal {V}-enriched categories, but we only treat the simple case - of Homset-categories here. - 360subb-000Esubb-000E.xmlI20245112024511318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson He#277 - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A( a , b )×A(F x , a ) \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] & X(U a , U b )×X( x , U a ) \ar [d, "∘"] \\ - A(F x , b ) \ar [r, "π_{ x , b }"] & X( x , U b ) - \end {tikzcd} - -Johnson Hehejohns-000G362subbb-000Esubbb-000E.xmlII20245112024511314Definitionsub-000Gsub-000G.xmlTopos à la Johnson He#277 - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - X( y , U a )×X( x , y ) \ar [r, "π_{ y , a }⁻¹× \overline {F} "] \ar [d, "∘"] & A(F y , a )×A(F x , F y ) \ar [d, "∘"] \\ - X( x , U a ) \ar [r, "π_{x, a}⁻¹"] & A(F x , a ) - \end {tikzcd} - -718Proof#278unstable-278.xml20245112024511Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson Hehejohns-000E - We use the same notation and definitions of η, ϵ, \overline {F} , \overline {U} as the - explicit proof of the precursor lemma. - The definition of \overline {F} , \overline {U} in the short proof - presupposes that U is a functor, so we're scrupulous to avoid - circularity. - - This proof is rather low tech and is lifted straight from the - paper. It's also how you'd attack the - precursor head-on. The utility is that it holds in - enriched categories. - - The forward directions are trivial. - Suppose (F, U, (π_{a, x})_{a ∈ A, x ∈ X}) extends to \overline {F} ⊣_π \overline {U}. - Then I expresses that ∀g : A(a, b). ∀f : A(F(x), a). π_{x, b}(g∘f) = \overline {U} (g)∘π_{a, x}(f). This is exactly the condition - that ∀g : A(a, b). π is natural in a at g, which is certainly - true of an adjunction. - II expresses something similar (dual). - - The nontrivial directions are the backward directions. - - (I) wts I ⟹ \overline {U} is a functor. - Suppose ∀a,b ∈ A. ∀x ∈ X. I holds. - Then in particular, ∀c ∈ A. I holds at x = U(c). - - - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A( a , b )×A(F Uc , a ) \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] & X(U a , U b )×X( Uc , U a ) \ar [d, "∘"] \\ - A(F Uc , b ) \ar [r, "π_{ Uc , b }"] & X( Uc , U b ) - \end {tikzcd} - - - - As does this trivial square: - - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\ - A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b) - \end {tikzcd} - - - which when pasted together, tells us magically: - - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(a, b)×A(c, a) \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\ - A(c, b) \ar [r, " \overline {U} "] & X(Uc, Ub) - \end {tikzcd} - - - Combined with the earlier proof that \overline {U} (1ₐ) = 1_{ \overline {U} (a)} (which just follows by definition), - we've shown that \overline {U} defines a functorial action on morphisms. - Actually, we saw earlier that I expresses that - π is natural in a, so we've met the preconditions of the - precursor lemma, thus there is a unique - extension \overline {F} st \overline {F} ⊣_π \overline {U}. - - - There's two ways ways to see the uniqueness situation: - +addr="hejohns">Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties - (An intermediate notion) \overline {U} is the unique extension of - U to a functor st π is natural in a. - We've essentially seen this already, if not literally. - For any functor U', naturality in a requires the following square to hold for all g : A(a, b): - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\ - X(U'a, U'a) \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] & X(U'a, U'b) - \end {tikzcd} - - Thus, ∀g. U'(g) = \overline {U} (g). - Then, with \overline {U} a functor and π natural in a, the - precursor gives us a unique \overline {F} st \overline {F} ⊣ \overline {U} uniquely. + (i) \mathcal {E} has all finite limits and colimits, - (Going backwards) Since the definitions of \overline {F} and - \overline {U} (in the precursor) are not dependent - on the other functor, the total adjunction \overline {F} ⊣ \overline {U} is - the unique one extending the core by applying the - uniqueness result of the precusor to each - functor in turn, just knowing the other is a well-defined - functor. - - - (II) wts II ⟹ \overline {F} is a functor. - This is dual. - \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω720subb-000Esubb-000E.xmlUntitled › I20245112024511Johnson He#277 - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A( a , b )×A(F x , a ) \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] & X(U a , U b )×X( x , U a ) \ar [d, "∘"] \\ - A(F x , b ) \ar [r, "π_{ x , b }"] & X( x , U b ) - \end {tikzcd} - -316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +722subbb-000Esubbb-000E.xmlUntitled › II20245112024511320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Johnson He#277 - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - X( y , U a )×X( x , y ) \ar [r, "π_{ y , a }⁻¹× \overline {F} "] \ar [d, "∘"] & A(F y , a )×A(F x , F y ) \ar [d, "∘"] \\ - X( x , U a ) \ar [r, "π_{x, a}⁻¹"] & A(F x , a ) - \end {tikzcd} - -Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +724Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He - Mentioned in passing as 2.1. Proposition., as "well-known". -Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +338Proposition#267unstable-267.xml2024572024511Johnson Hehejohns-0007 - Suppose U : A → X is a functor, F : X .ob → A .ob a function, - and (πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X} a family of natural - isomorphisms. Then there is a unique extension \overline {F} of F such - that (πₓ)_{x ∈ X} is natural in x, witnessing \overline {F} ⊣ U. - - - Or dually, suppose that F is a functor, but U is just a - function on objects, and (πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}. - Then U uniquely extends to \overline {U}. - - Let's be pedantic and clarify that A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ - have the same action on morphisms as Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U. - This is conventionally left implicit, but we remark to clarify what we mean - by the family of natural isomorphisms being natural. - - 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. -327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 - Suppose such a \overline {F} exists. - Then naturality of π requires that - ∀f : X(x, y). this square in Setᴬ commutes: - 325#269unstable-269.xmlSyntax202463202463hejohns-000J340subsub-0007subsub-0007.xml2024572024511326#270unstable-270.xmlSemantics202463202463hejohns-000J909#278unstable-278.xmlElementary² toposes › elementary questions2024531202463Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - Then clearly ∃! \overline {F} .∀f : X(x, y). \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y). - - - - (dual) Suppose such a \overline {U} exists. - Then naturality of π requires that - ∀u : A(a, b). this square in Set^{Xᵒᵖ} commutes: - Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 342subsubb-0007subsubb-0007.xml2024572024511911Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(-), a) \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\ - X(-, U(a)) \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] & X(-, U(b)) - \end {tikzcd} - - and clearly ∃! \overline {U} .∀u : A(a, b). \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹). - Johnson He354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511330#278unstable-278.xmlElementary questions2024531202463Johnson He - We take inspiration from Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following the thin case. - We wts the F-action on morphisms is fully determined already (∃!). - That is, we have two goals: - - (∃) Show that there is a suitable F-action. - - - (!) That any other F-action is equal to it. - - As usual, let η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x)) denote the - component of the unit of the (a posteriori) adjunction - and ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)}) denote the component of the counit. - - (∃) - - Now suppose ∃f : X(x, y). - Since η_y : X(y, U∘F(y)), we have η_y∘f : X(x, U∘F(y)). - By the Hom bijection, πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y)). - This is our candidate for the F-action. - \overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f). - - - As a picture, each f : X(x, y) gives rise to the following - square in Set: - Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 346subb-000Dsubb-000D.xml20245102024511332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\ - X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"] - \end {tikzcd} - - And our candidate \overline {F} (f) is the one we know to exist by - chasing 1_{F(y)} around the square. - - - We could directly try to show that \overline {F} defined as such is - indeed functorial--and I have tried--, but it's rather - difficult without the clarifying perspective of the Yoneda - lemma (no, seriously, although I'm sure one could do it). - - - It's instructive to try directly, if one hasn't. - The problem is that \overline {F} (f) as defined is not obviously the - composition of morphisms in any diagram-- how do you reason - about the functoriality of πₓ(F(y))⁻¹? - Instead, we want to invoke a representation theorem and work - with \overline {F} (f) as an explicit composition in some diagram. - The solution is to follow the Yoneda principle and - pass from reasoning in Set to Setᴬ. - - - Notice that the above square is - moreover natural in the variable F(y). - That is, we've actually described the F(y) component of the - natural transformation πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y), -). - As a picture, each f : X(x, y) gives - rise to the following square in Setᴬ: - Johnson Hehejohns-000C348sub-000Dsub-000D.xml20245102024511302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\ - X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - - We first make the trivial observation that A(F(x), -) and - A(F(y), -) are both Yoneda embeddings (we use the "version" - where よ : Aᵒᵖ → Setᴬ, ie the Yoneda embedding on Aᵒᵖ), - よ(F(x)) and よ(F(y)) respectively. - - - Then the Yoneda lemma tells us that by specifying an - element \overline {F} (f) : A(F(x), F(y)), this fully - determines (ie freely extends to) a natural - transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) - - Equivalently, the Yoneda principle tells us that an - element \overline {F} (f) : A(F(x), F(y)) fully failthfully - embeds as a natural transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) - - With this in mind, we notice that by definition, - よ( \overline {F} (f)) is exactly the lid to Johnson Hethe above Setᴬ - square. - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - That is, both natural transformations map 1_{F(y)} to the - same thing-- \overline {F} {F}(f)-- by definition, thus must be equal - by the Yoneda Lemma. - (Topoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If In hindsight, this line of reasoning is quite - circumlocutious and you might as well start by saying such a - \overline {F} 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 \overline {F} (f) as an explicit - composition, we have enough to tools consider - functoriality. - - - Wts \overline {F} actually defines a functorial action on morphisms for a functor extending F. - Indeed, \overline {F} (1ₓ : X(x, x)) = πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)} by definition. - And suppose ∃g : X(y, z). - Then \overline {F} (g∘f : X(x, z)) viewed as a natural transformation gives rise to the following superimposed squares: - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] & 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"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left] - \end {tikzcd} - - A diagram chase verifies that the diagram indeed commutes, and - that よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g)). - By the ("going home" direction of the) Yoneda principle, \overline {F} (g∘f) = \overline {F} (f)∘ \overline {F} (g). - Thus at last, \overline {F} defines a functorial action on X-morphisms. - - - We should also check that \overline {F} is actually adjoint to U-- - ie, that (πₓ)_{x ∈ X} is indeed natural in the variable - x as well as a. - By great good fortune, by defining \overline {F} st - ∀x,y ∈ X. ∀f ∈ X(x, y). - Quantifiers and Sheaves 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. +350subsub-0007subsub-0007.xml2024572024511321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - it is indeed the case. - - - (!) - - To finish off the result, we wts for any other F' extending - F to morphisms st F' ⊣ U, witnessed by π : Xᵒᵖ → Setᴬ, - that F' = \overline {F}. - This simply follows from the observation that ∀f. naturality - of π is exactly this square, and forces - よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f)) - So this result is really a corollary of the Yoneda lemma. - The nlab page is here. +addr="hejohns">Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. 352Moralsubbb-000Dsubbb-000D.xml20245102024511304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000D - Of course in hindsight, we could've (or even should've) - just defined \overline {F} to be - よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y) 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. - - This is the perspective taken in the short proof. - - However, unlike the thin case, - we crucially use the information that U is a functor - so the natural transformation squares are well-defined. -Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 726Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson HeJohnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 358Definition#276unstable-276.xml20245112024511308Definition#273unstable-273.xmlTopos à la Johnson Hehejohns-000ESketches of an Elephant: a topos theory compendium 2.1.1202461202463"An adjunction core consists of" - - functions U : A .ob → X .ob and F : X .ob → A .ob - - a family of isomorphisms (π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 364Proposition#277unstable-277.xml20245112024511Johnson Hehejohns-000E - An adjunction core extends to an adjuction iff ∀a,b ∈ A. ∀x ∈ X. - I commutes, and iff ∀x,y ∈ X. a ∈ A. 310Definition#274unstable-274.xmlTopos à la II - commutes. - The adjunction is unique when it exists. - - The Category Theory 8.17202461202463original theorem is stated in terms of - \mathcal {V}-enriched categories, but we only treat the simple case - of Homset-categories here. - Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 360subb-000Esubb-000E.xmlI20245112024511312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson He#277 - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A( a , b )×A(F x , a ) \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] & X(U a , U b )×X( x , U a ) \ar [d, "∘"] \\ - A(F x , b ) \ar [r, "π_{ x , b }"] & X( x , U b ) - \end {tikzcd} - -Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 362subbb-000Esubbb-000E.xmlII20245112024511318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson He#277 - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - X( y , U a )×X( x , y ) \ar [r, "π_{ y , a }⁻¹× \overline {F} "] \ar [d, "∘"] & A(F y , a )×A(F x , F y ) \ar [d, "∘"] \\ - X( x , U a ) \ar [r, "π_{x, a}⁻¹"] & A(F x , a ) - \end {tikzcd} - -Johnson Hehejohns-000G366Proof#278unstable-278.xml20245112024511Johnson Hehejohns-000E - We use the same notation and definitions of η, ϵ, \overline {F} , \overline {U} as the - explicit proof of the precursor lemma. - The definition of \overline {F} , \overline {U} in the short proof - presupposes that U is a functor, so we're scrupulous to avoid - circularity. - - This proof is rather low tech and is lifted straight from the - paper. It's also how you'd attack the - precursor head-on. The utility is that it holds in - enriched categories. - - The forward directions are trivial. - Suppose (F, U, (π_{a, x})_{a ∈ A, x ∈ X}) extends to \overline {F} ⊣_π \overline {U}. - Then I expresses that ∀g : A(a, b). ∀f : A(F(x), a). π_{x, b}(g∘f) = \overline {U} (g)∘π_{a, x}(f). This is exactly the condition - that ∀g : A(a, b). π is natural in a at g, which is certainly - true of an adjunction. - II expresses something similar (dual). - - The nontrivial directions are the backward directions. - - (I) wts I ⟹ \overline {U} is a functor. - Suppose ∀a,b ∈ A. ∀x ∈ X. I holds. - Then in particular, ∀c ∈ A. I holds at x = U(c). - - - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A( a , b )×A(F Uc , a ) \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] & X(U a , U b )×X( Uc , U a ) \ar [d, "∘"] \\ - A(F Uc , b ) \ar [r, "π_{ Uc , b }"] & X( Uc , U b ) - \end {tikzcd} - - - - As does this trivial square: - - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\ - A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b) - \end {tikzcd} - - - which when pasted together, tells us magically: - - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(a, b)×A(c, a) \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\ - A(c, b) \ar [r, " \overline {U} "] & X(Uc, Ub) - \end {tikzcd} - - - Combined with the earlier proof that \overline {U} (1ₐ) = 1_{ \overline {U} (a)} (which just follows by definition), - we've shown that \overline {U} defines a functorial action on morphisms. - Actually, we saw earlier that 314Definitionsub-000Gsub-000G.xmlTopos à la I expresses that - π is natural in a, so we've met the preconditions of Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463the - precursor lemma, thus there is a unique - extension \overline {F} st \overline {F} ⊣_π \overline {U}. - - - There's two ways ways to see the uniqueness situation: - +addr="hejohns">Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties - (An intermediate notion) \overline {U} is the unique extension of - U to a functor st π is natural in a. - We've essentially seen this already, if not literally. - For any functor U', naturality in a requires the following square to hold for all g : A(a, b): - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\ - X(U'a, U'a) \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] & X(U'a, U'b) - \end {tikzcd} - - Thus, ∀g. U'(g) = \overline {U} (g). - Then, with \overline {U} a functor and π natural in a, the - precursor gives us a unique \overline {F} st \overline {F} ⊣ \overline {U} uniquely. + (i) \mathcal {E} has all finite limits and colimits, - (Going backwards) Since the definitions of \overline {F} and - \overline {U} (in the precursor) are not dependent - on the other functor, the total adjunction \overline {F} ⊣ \overline {U} is - the unique one extending the core by applying the - uniqueness result of the precusor to each - functor in turn, just knowing the other is a well-defined - functor. - - - (II) wts II ⟹ \overline {F} is a functor. - This is dual. - \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω368Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E - (For V-enriched categories, ) If V is thin, all diagrams commute, - thus an adjunction core always determines an adjunction. - Set isn't thin, but 2 is, thus for posets, an adjunction core - gives rise to a thin adjunction, as remarked 316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from here. - I don't know how useful this lemma is for ordinary Homset categories. - Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +728Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He - This is mentioned in passing, in the intro to The Core of Adjoint Functors. -Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +330Proposition#268unstable-268.xml2024572024511Johnson Hehejohns-0006 - Suppose P and Q are posets, f : P → Q is a priori just a function, and u : P ← Q monotone, - such that f(x) ≤ a ⟺ x ≤ u(a). - Then f monotone as well. - - Or dually, suppose that f monotone, but u is a priori just a function. - Then u monotone as well. - 327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463332Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 - Suppose p₁ ≤ p₂. - Notice that p₂ ≤ u∘f(p₂), by the unit of the (thin) adjunction. - Then p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂) by the Hom bijection of the (thin) adjunction. - - (dual) - Suppose q₁ ≤ q₂. - Notice that f∘u(q₁) ≤ q₂ by the counit of the (thin) adjunction. - Then f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂) by the Hom bijection of the (thin) adjunction. - 325#269unstable-269.xmlSyntax202463202463hejohns-000J334Proof corollarysubb-0006subb-0006.xml2024572024511326#270unstable-270.xmlSemantics202463202463hejohns-000J913hejohns-000Ahejohns-000A.xmlFiberwise Structure202458202461Johnson Hehejohns-0006 - Notice that the above proof does not use the information that u is monotone. - Thus Johnson He + Fibered Right Adjoints are automatically fibered? +915hejohns-000Bhejohns-000B.xmlLogical Frameworks as a meta-language2024527202461the above lemma holds where f and u are just a priori functions. - The Johnson He + While reading above lemma is in some sense the more - immediately natural question, but can be easily strengthened to - the form of this corollary. - An Axiomatization of Set Theory, 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. + + That is, (classical) FoL is often the setting in which axiomatic + mathematical theories are formalized. Classic (no pun intended) examples + are ( \mathcal {L}_{group} = \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups})), ( \mathcal {L}_{DLO} = \{ ≤ \} , Th( \textsf {Dense Linear Orders})), ( \mathcal {L}_{sets} = \{ ∈ \} , ZFC). + (To be ultra-pedantic is a fruitless way, I've paired the language and theory.) + + "definite property" +730Proofsub-0007sub-0007.xml917#259unstable-259.xmlPrecursor to The Core of Adjoint Functors › the short proof (sketch)2024572024511Functors are Type Refinement Systems › Sections 1-22024511202461Johnson Hehejohns-0007 - Suppose such a \overline {F} exists. - Then naturality of π requires that - ∀f : X(x, y). this square in Setᴬ commutes: - Johnson Hemillies-zeilberger-2015What is a logical relation? + A LR is just a model of the syntax, used to prove ∀M statements. + Thus, it's usually just 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 just a clever choice of model, + to let us say something interesting/talk about what we care about. + + 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 two 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. + + This way, we can show that two different semantics and implementations + for a intset 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. + + 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? + + 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. + + 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? + + Maybe I'm finally close to being able to understand what a Kripke + logical relation is. + 340subsub-0007subsub-0007.xml2024572024511919hejohns-0009hejohns-0009.xmlFibrations v. functors202458202461Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - Then clearly ∃! \overline {F} .∀f : X(x, y). \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y). - - - - (dual) Suppose such a \overline {U} exists. - Then naturality of π requires that - ∀u : A(a, b). this square in Set^{Xᵒᵖ} commutes: - Johnson He + What is the actual logic when doing a logical relation? + Do you only actually need functors? Zeilberger's Functors as Type Refinement Systems, is a must read. + Then you lose the internal logic... maybe? Maybe you just have a different internal logic? + + 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. + paper342subsubb-0007subsubb-0007.xml2024572024511921Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(-), a) \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\ - X(-, U(a)) \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] & X(-, U(b)) - \end {tikzcd} - - and clearly ∃! \overline {U} .∀u : A(a, b). \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹). - Johnson HeThis is a fundamental notion. + 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 notes on + realizability, + according to Loris it's relevant to Markov Decision Processes, and we + develop the view that it's a logician's view of fibrations. +732Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511521Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He - We take inspiration from + The concept first presented is that of 517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456the thin case. - We wts the F-action on morphisms is fully determined already (∃!). - That is, we have two goals: - - (∃) Show that there is a suitable F-action. +addr="hejohns">Johnson He + "The basic case": + + A relation Z (to not be confused with the accessibility relations R for + each Kripke frame) between two Kripke frames M and M' is a *bisimulation* iff + w Z w' ⟹ (w and w' are elementarily equivalent) - (!) That any other F-action is equal to it. - - As usual, let η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x)) denote the - component of the unit of the (a posteriori) adjunction - and ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)}) denote the component of the counit. - - (∃) - - Now suppose ∃f : X(x, y). - Since η_y : X(y, U∘F(y)), we have η_y∘f : X(x, U∘F(y)). - By the Hom bijection, πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y)). - This is our candidate for the F-action. - \overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f). - - - As a picture, each f : X(x, y) gives rise to the following - square in Set: - forth) w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v' + + (backwards) w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'346subb-000Dsubb-000D.xml20245102024511515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\ - X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"] - \end {tikzcd} - - And our candidate \overline {F} (f) is the one we know to exist by - chasing 1_{F(y)} around the square. - - - We could directly try to show that \overline {F} defined as such is - indeed functorial--and I have tried--, but it's rather - difficult without the clarifying perspective of the Yoneda - lemma (no, seriously, although I'm sure one could do it). - - - It's instructive to try directly, if one hasn't. - The problem is that \overline {F} (f) as defined is not obviously the - composition of morphisms in any diagram-- how do you reason - about the functoriality of πₓ(F(y))⁻¹? - Instead, we want to invoke a representation theorem and work - with \overline {F} (f) as an explicit composition in some diagram. - The solution is to follow the Yoneda principle and - pass from reasoning in Set to Setᴬ. - - - Notice that the above square is - moreover natural in the variable F(y). - That is, we've actually described the F(y) component of the - natural transformation πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y), -). - As a picture, each f : X(x, y) gives - rise to the following square in Setᴬ: - Johnson He + "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" + +. + A simulation is where we drop the backwards condition. +348sub-000Dsub-000D.xml20245102024511519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\ - X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - - We first make the trivial observation that A(F(x), -) and - A(F(y), -) are both Yoneda embeddings (we use the "version" - where よ : Aᵒᵖ → Setᴬ, ie the Yoneda embedding on Aᵒᵖ), - よ(F(x)) and よ(F(y)) respectively. - - - Then the Yoneda lemma tells us that by specifying an - element \overline {F} (f) : A(F(x), F(y)), this fully - determines (ie freely extends to) a natural - transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) - - Equivalently, the Yoneda principle tells us that an - element \overline {F} (f) : A(F(x), F(y)) fully failthfully - embeds as a natural transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) - - With this in mind, we notice that by definition, - よ( \overline {F} (f)) is exactly the lid to the above Setᴬ - square. - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - That is, both natural transformations map 1_{F(y)} to the - same thing-- \overline {F} {F}(f)-- by definition, thus must be equal - by the Yoneda Lemma. - (In hindsight, this line of reasoning is quite - circumlocutious and you might as well start by saying such a - \overline {F} 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 \overline {F} (f) as an explicit - composition, we have enough to tools consider - functoriality. - - - Wts \overline {F} actually defines a functorial action on morphisms for a functor extending F. - Indeed, \overline {F} (1ₓ : X(x, x)) = πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)} by definition. - And suppose ∃g : X(y, z). - Then \overline {F} (g∘f : X(x, z)) viewed as a natural transformation gives rise to the following superimposed squares: - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] & 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"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left] - \end {tikzcd} - - A diagram chase verifies that the diagram indeed commutes, and - that よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g)). - By the ("going home" direction of the) Yoneda principle, \overline {F} (g∘f) = \overline {F} (f)∘ \overline {F} (g). - Thus at last, \overline {F} defines a functorial action on X-morphisms. - - - We should also check that \overline {F} is actually adjoint to U-- - ie, that (πₓ)_{x ∈ X} is indeed natural in the variable - x as well as a. - By great good fortune, by defining \overline {F} st - ∀x,y ∈ X. ∀f ∈ X(x, y). - Johnson He + Clearly the truth of ∧,∨,◇-formulas are preserved under simulations. + Actually it's shown that the class of formulas preserved under simulations are exactly these formulas. +350subsub-0007subsub-0007.xml2024572024511923Email#257unstable-257.xml20245282024529Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - it is indeed the case. - - - (!) - - To finish off the result, we wts for any other F' extending - F to morphisms st F' ⊣ U, witnessed by π : Xᵒᵖ → Setᴬ, - that F' = \overline {F}. - This simply follows from the observation that ∀f. naturality - of π is exactly this square, and forces - よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f)) - So this result is really a corollary of the Yoneda lemma. - The nlab page is here. -Johnson Hequine-1960 + Thanks for sharing this nice little note, which sketches a combinatory + (classical) first-order logic. + + I was surprised by the mention of von Neumann 1925 in this snippet: + + "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) + + 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. + + 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 class existence theorem. + + (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.) + + 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. + + 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. + + 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, + + Johnson He + 352Moralsubbb-000Dsubbb-000D.xml20245102024511925#283unstable-283.xmlThe Logical Basis of Metaphysics › Stability20245122024513Johnson Hehejohns-000D - Of course in hindsight, we could've (or even should've) - just defined \overline {F} to be - よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y) 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 Hedummett-91 + The opening sentence is what zeilberger quotes on his work on + polarity. - This is the perspective taken in the short proof. - 734Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510927Referencedummett-91dummett-91.xmlThe Logical Basis of Metaphysics20245122024513Johnson He336Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511599#283unstable-283.xmlStability20245122024513Johnson He - This is mentioned in passing, in the intro to The Core of Adjoint Functors. -Johnson Hedummett-91 + The opening sentence is what zeilberger quotes on his work on + polarity. + + 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. + 330Proposition#268unstable-268.xml2024572024511929Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 - Suppose P and Q are posets, f : P → Q is a priori just a function, and u : P ← Q monotone, - such that f(x) ≤ a ⟺ x ≤ u(a). - Then f monotone as well. + Suppose p₁ ≤ p₂. + Notice that p₂ ≤ u∘f(p₂), by the unit of the (thin) adjunction. + Then p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂) by the Hom bijection of the (thin) adjunction. - Or dually, suppose that f monotone, but u is a priori just a function. - Then u monotone as well. + (dual) + Suppose q₁ ≤ q₂. + Notice that f∘u(q₁) ≤ q₂ by the counit of the (thin) adjunction. + Then f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂) by the Hom bijection of the (thin) adjunction. 332Proofsub-0006sub-0006.xml2024572024511931sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-0006 - Suppose p₁ ≤ p₂. - Notice that p₂ ≤ u∘f(p₂), by the unit of the (thin) adjunction. - Then p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂) by the Hom bijection of the (thin) adjunction. - - (dual) - Suppose q₁ ≤ q₂. - Notice that f∘u(q₁) ≤ q₂ by the counit of the (thin) adjunction. - Then f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂) by the Hom bijection of the (thin) adjunction. +addr="hejohns">Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\ + X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + 933Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E + (For V-enriched categories, ) If V is thin, all diagrams commute, + thus an adjunction core always determines an adjunction. + Set isn't thin, but 2 is, thus for posets, an adjunction core + gives rise to a thin adjunction, as remarked here. + I don't know how useful this lemma is for ordinary Homset categories. 334Proof corollarysubb-0006subb-0006.xml2024572024511935Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -1864,71 +2021,54 @@ type="local" addr="hejohns-0006">above lemma is in some sense the more immediately natural question, but can be easily strengthened to the form of this corollary. - 356Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511937subb-000Dsubb-000D.xml20245102024511Johnson He - Mentioned in passing as 2.1. Proposition., as "well-known". -Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\ + X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"] + \end {tikzcd} + 338Proposition#267unstable-267.xml2024572024511939Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-0007 - Suppose U : A → X is a functor, F : X .ob → A .ob a function, - and (πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X} a family of natural - isomorphisms. Then there is a unique extension \overline {F} of F such - that (πₓ)_{x ∈ X} is natural in x, witnessing \overline {F} ⊣ U. - - - Or dually, suppose that F is a functor, but U is just a - function on objects, and (πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}. - Then U uniquely extends to \overline {U}. - - Let's be pedantic and clarify that A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ - have the same action on morphisms as Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U. - This is conventionally left implicit, but we remark to clarify what we mean - by the family of natural isomorphisms being natural. - - 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. -344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-000D + Of course in hindsight, we could've (or even should've) + just defined \overline {F} to be + よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y) 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. + + This is the perspective taken in Johnson Hehejohns-0007 - Suppose such a \overline {F} exists. - Then naturality of π requires that - ∀f : X(x, y). this square in Setᴬ commutes: - the short proof. + 340subsub-0007subsub-0007.xml2024572024511941subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007\usepackage {t \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] \end {tikzcd} - - Then clearly ∃! \overline {F} .∀f : X(x, y). \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y). - - - - (dual) Suppose such a \overline {U} exists. - Then naturality of π requires that - ∀u : A(a, b). this square in Set^{Xᵒᵖ} commutes: - 342subsubb-0007subsubb-0007.xml2024572024511943subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007\usepackage {t A(F(-), a) \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\ X(-, U(a)) \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] & X(-, U(b)) \end {tikzcd} - - and clearly ∃! \overline {U} .∀u : A(a, b). \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹). - 354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511945Proposition#280unstable-280.xml2024572024511Johnson He - We take inspiration from the thin case. - We wts the F-action on morphisms is fully determined already (∃!). - That is, we have two goals: - - (∃) Show that there is a suitable F-action. +addr="hejohns">Johnson Hehejohns-0007 + Suppose U : A → X is a functor, F : X .ob → A .ob a function, + and (πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X} a family of natural + isomorphisms. Then there is a unique extension \overline {F} of F such + that (πₓ)_{x ∈ X} is natural in x, witnessing \overline {F} ⊣ U. - (!) That any other F-action is equal to it. - - As usual, let η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x)) denote the - component of the unit of the (a posteriori) adjunction - and ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)}) denote the component of the counit. - - (∃) - - Now suppose ∃f : X(x, y). - Since η_y : X(y, U∘F(y)), we have η_y∘f : X(x, U∘F(y)). - By the Hom bijection, πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y)). - This is our candidate for the F-action. - \overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f). - - - As a picture, each f : X(x, y) gives rise to the following - square in Set: - F is a functor, but U is just a + function on objects, and (πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}. + Then U uniquely extends to \overline {U}. + 346subb-000Dsubb-000D.xml20245102024511947Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\ - X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"] - \end {tikzcd} - - And our candidate \overline {F} (f) is the one we know to exist by - chasing 1_{F(y)} around the square. - - - We could directly try to show that \overline {F} defined as such is - indeed functorial--and I have tried--, but it's rather - difficult without the clarifying perspective of the Yoneda - lemma (no, seriously, although I'm sure one could do it). - - - It's instructive to try directly, if one hasn't. - The problem is that \overline {F} (f) as defined is not obviously the - composition of morphisms in any diagram-- how do you reason - about the functoriality of πₓ(F(y))⁻¹? - Instead, we want to invoke a representation theorem and work - with \overline {F} (f) as an explicit composition in some diagram. - The solution is to follow the Yoneda principle and - pass from reasoning in Set to Setᴬ. - - - Notice that the above square is - moreover natural in the variable F(y). - That is, we've actually described the F(y) component of the - natural transformation πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y), -). - As a picture, each f : X(x, y) gives - rise to the following square in Setᴬ: - Johnson Hehejohns-0006 + Suppose P and Q are posets, f : P → Q is a priori just a function, and u : P ← Q monotone, + such that f(x) ≤ a ⟺ x ≤ u(a). + Then f monotone as well. + + Or dually, suppose that f monotone, but u is a priori just a function. + Then u monotone as well. + 348sub-000Dsub-000D.xml20245102024511949Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\ - X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - - We first make the trivial observation that A(F(x), -) and - A(F(y), -) are both Yoneda embeddings (we use the "version" - where よ : Aᵒᵖ → Setᴬ, ie the Yoneda embedding on Aᵒᵖ), - よ(F(x)) and よ(F(y)) respectively. - - - Then the Yoneda lemma tells us that by specifying an - element \overline {F} (f) : A(F(x), F(y)), this fully - determines (ie freely extends to) a natural - transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) - - Equivalently, the Yoneda principle tells us that an - element \overline {F} (f) : A(F(x), F(y)) fully failthfully - embeds as a natural transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) - - With this in mind, we notice that by definition, - よ( \overline {F} (f)) is exactly the lid to the above Setᴬ - square. - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - That is, both natural transformations map 1_{F(y)} to the - same thing-- \overline {F} {F}(f)-- by definition, thus must be equal - by the Yoneda Lemma. - (In hindsight, this line of reasoning is quite - circumlocutious and you might as well start by saying such a - \overline {F} 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 \overline {F} (f) as an explicit - composition, we have enough to tools consider - functoriality. - - - Wts \overline {F} actually defines a functorial action on morphisms for a functor extending F. - Indeed, \overline {F} (1ₓ : X(x, x)) = πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)} by definition. - And suppose ∃g : X(y, z). - Then \overline {F} (g∘f : X(x, z)) viewed as a natural transformation gives rise to the following superimposed squares: - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] & 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"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left] - \end {tikzcd} - - A diagram chase verifies that the diagram indeed commutes, and - that よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g)). - By the ("going home" direction of the) Yoneda principle, \overline {F} (g∘f) = \overline {F} (f)∘ \overline {F} (g). - Thus at last, \overline {F} defines a functorial action on X-morphisms. - - - We should also check that \overline {F} is actually adjoint to U-- - ie, that (πₓ)_{x ∈ X} is indeed natural in the variable - x as well as a. - By great good fortune, by defining \overline {F} st - ∀x,y ∈ X. ∀f ∈ X(x, y). - 350subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ - X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] - \end {tikzcd} - - it is indeed the case. - - - (!) - - To finish off the result, we wts for any other F' extending - F to morphisms st F' ⊣ U, witnessed by π : Xᵒᵖ → Setᴬ, - that F' = \overline {F}. - This simply follows from the observation that ∀f. naturality - of π is exactly this square, and forces - よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f)) - So this result is really a corollary of the Yoneda lemma. - The nlab page is here. -352Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D - Of course in hindsight, we could've (or even should've) - just defined \overline {F} to be - よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y) 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. - - This is the perspective taken in the short proof. - - However, unlike the thin case, - we crucially use the information that U is a functor - so the natural transformation squares are well-defined. -370Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511Johnson Hehejohns-000EJohnson Hehejohns-000E"An adjunction core consists of" @@ -2224,11 +2149,11 @@ addr="street-2012">"An adjunction core consists of" a family of isomorphisms (π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}364Proposition#277unstable-277.xml20245112024511951Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -2256,10 +2181,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -2274,10 +2199,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -2288,11 +2213,11 @@ hash="8fd7996c4b1a3e39d4bda43697f06cf9">\usepackage {t 366Proof#278unstable-278.xml20245112024511953Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -2427,373 +2352,632 @@ addr="hejohns-0007">the precusor to each 368Proof corollarysub-000Esub-000E.xml20245112024511955subb-000Esubb-000E.xmlJohnson Hehejohns-000E - (For V-enriched categories, ) If V is thin, all diagrams commute, - thus an adjunction core always determines an adjunction. - Set isn't thin, but 2 is, thus for posets, an adjunction core - gives rise to a thin adjunction, as remarked Untitled › I20245112024511here. - I don't know how useful this lemma is for ordinary Homset categories. - Johnson He#290 + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A( a , b )×A(F x , a ) \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] & X(U a , U b )×X( x , U a ) \ar [d, "∘"] \\ + A(F x , b ) \ar [r, "π_{ x , b }"] & X( x , U b ) + \end {tikzcd} + +736Referencejacobs-1999jacobs-1999.xmlCategorical Logic and Type Theory202456202456957subbb-000Esubbb-000E.xmlUntitled › II20245112024511Johnson HeJohnson He#290 + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + X( y , U a )×X( x , y ) \ar [r, "π_{ y , a }⁻¹× \overline {F} "] \ar [d, "∘"] & A(F y , a )×A(F x , F y ) \ar [d, "∘"] \\ + X( x , U a ) \ar [r, "π_{x, a}⁻¹"] & A(F x , a ) + \end {tikzcd} + +738Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456959Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He - "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" - -2.1. Proposition., as "well-known". +740Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456533Proposition#280unstable-280.xml2024572024511Johnson He - "The basic case": - - A relation Z (to not be confused with the accessibility relations R for - each Kripke frame) between two Kripke frames M and M' is a *bisimulation* iff - w Z w' ⟹ (w and w' are elementarily equivalent) +addr="hejohns">Johnson Hehejohns-0007 + Suppose U : A → X is a functor, F : X .ob → A .ob a function, + and (πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X} a family of natural + isomorphisms. Then there is a unique extension \overline {F} of F such + that (πₓ)_{x ∈ X} is natural in x, witnessing \overline {F} ⊣ U. - (forth) w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v' - - (backwards) w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'F is a functor, but U is just a + function on objects, and (πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}. + Then U uniquely extends to \overline {U}. + + Let's be pedantic and clarify that A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ + have the same action on morphisms as Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U. + This is conventionally left implicit, but we remark to clarify what we mean + by the family of natural isomorphisms being natural. + + 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. +320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson He - "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" - -Johnson Hehejohns-0007 + Suppose such a \overline {F} exists. + Then naturality of π requires that + ∀f : X(x, y). this square in Setᴬ commutes: + 742Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456535subsub-0007subsub-0007.xml2024572024511Johnson He - Clearly the truth of ∧,∨,◇-formulas are preserved under simulations. - Actually it's shown that the class of formulas preserved under simulations are exactly these formulas. -Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + Then clearly ∃! \overline {F} .∀f : X(x, y). \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y). + + + + (dual) Suppose such a \overline {U} exists. + Then naturality of π requires that + ∀u : A(a, b). this square in Set^{Xᵒᵖ} commutes: + 744Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456537subsubb-0007subsubb-0007.xml2024572024511Johnson He - Here is Blackburn et all's account: - - The concept first presented is that of Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(-), a) \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\ + X(-, U(a)) \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] & X(-, U(b)) + \end {tikzcd} + + and clearly ∃! \overline {U} .∀u : A(a, b). \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹). + 322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He - "The basic case": - - A relation Z (to not be confused with the accessibility relations R for - each Kripke frame) between two Kripke frames M and M' is a *bisimulation* iff - w Z w' ⟹ (w and w' are elementarily equivalent) + We take inspiration from the thin case. + We wts the F-action on morphisms is fully determined already (∃!). + That is, we have two goals: + + (∃) Show that there is a suitable F-action. - (forth) w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v' - - (backwards) w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'F-action is equal to it. + + As usual, let η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x)) denote the + component of the unit of the (a posteriori) adjunction + and ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)}) denote the component of the counit. + + (∃) + + Now suppose ∃f : X(x, y). + Since η_y : X(y, U∘F(y)), we have η_y∘f : X(x, U∘F(y)). + By the Hom bijection, πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y)). + This is our candidate for the F-action. + \overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f). + + + As a picture, each f : X(x, y) gives rise to the following + square in Set: + 320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456541subb-000Dsubb-000D.xml20245102024511Johnson He - "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" - -. - A simulation is where we drop the backwards condition. -Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\ + X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"] + \end {tikzcd} + + And our candidate \overline {F} (f) is the one we know to exist by + chasing 1_{F(y)} around the square. + + + We could directly try to show that \overline {F} defined as such is + indeed functorial--and I have tried--, but it's rather + difficult without the clarifying perspective of the Yoneda + lemma (no, seriously, although I'm sure one could do it). + + + It's instructive to try directly, if one hasn't. + The problem is that \overline {F} (f) as defined is not obviously the + composition of morphisms in any diagram-- how do you reason + about the functoriality of πₓ(F(y))⁻¹? + Instead, we want to invoke a representation theorem and work + with \overline {F} (f) as an explicit composition in some diagram. + The solution is to follow the Yoneda principle and + pass from reasoning in Set to Setᴬ. + + + Notice that the above square is + moreover natural in the variable F(y). + That is, we've actually described the F(y) component of the + natural transformation πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y), -). + As a picture, each f : X(x, y) gives + rise to the following square in Setᴬ: + 324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456543sub-000Dsub-000D.xml20245102024511Johnson He - Clearly the truth of ∧,∨,◇-formulas are preserved under simulations. - Actually it's shown that the class of formulas preserved under simulations are exactly these formulas. -675indexindex.xmlJohnson He202455202461false - - Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\ + X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + + We first make the trivial observation that A(F(x), -) and + A(F(y), -) are both Yoneda embeddings (we use the "version" + where よ : Aᵒᵖ → Setᴬ, ie the Yoneda embedding on Aᵒᵖ), + よ(F(x)) and よ(F(y)) respectively. + + + Then the Yoneda lemma tells us that by specifying an + element \overline {F} (f) : A(F(x), F(y)), this fully + determines (ie freely extends to) a natural + transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) + + Equivalently, the Yoneda principle tells us that an + element \overline {F} (f) : A(F(x), F(y)) fully failthfully + embeds as a natural transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) + + With this in mind, we notice that by definition, + よ( \overline {F} (f)) is exactly the lid to the above Setᴬ + square. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + That is, both natural transformations map 1_{F(y)} to the + same thing-- \overline {F} {F}(f)-- by definition, thus must be equal + by the Yoneda Lemma. + (In hindsight, this line of reasoning is quite + circumlocutious and you might as well start by saying such a + \overline {F} 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 \overline {F} (f) as an explicit + composition, we have enough to tools consider + functoriality. + + + Wts \overline {F} actually defines a functorial action on morphisms for a functor extending F. + Indeed, \overline {F} (1ₓ : X(x, x)) = πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)} by definition. + And suppose ∃g : X(y, z). + Then \overline {F} (g∘f : X(x, z)) viewed as a natural transformation gives rise to the following superimposed squares: + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] & 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"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left] + \end {tikzcd} + + A diagram chase verifies that the diagram indeed commutes, and + that よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g)). + By the ("going home" direction of the) Yoneda principle, \overline {F} (g∘f) = \overline {F} (f)∘ \overline {F} (g). + Thus at last, \overline {F} defines a functorial action on X-morphisms. + + + We should also check that \overline {F} is actually adjoint to U-- + ie, that (πₓ)_{x ∈ X} is indeed natural in the variable + x as well as a. + By great good fortune, by defining \overline {F} st + ∀x,y ∈ X. ∀f ∈ X(x, y). + 314Personhejohnshejohns.xmlAbout Me202455202459My screen name is hejohns. This is my forest. - email: hejohns@umich.edu -TODO: I've copied over what's on https://hejohns.github.io/, but this isn't a good selection.545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + it is indeed the case. + + + (!) + + To finish off the result, we wts for any other F' extending + F to morphisms st F' ⊣ U, witnessed by π : Xᵒᵖ → Setᴬ, + that F' = \overline {F}. + This simply follows from the observation that ∀f. naturality + of π is exactly this square, and forces + よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f)) + So this result is really a corollary of the Yoneda lemma. + The nlab page is here. +315#261unstable-261.xmlCode202455202461indexGradescope-Utils, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly - - misc cmdline utils. colorless is nice for long git status output (it's really color-less). - 547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D + Of course in hindsight, we could've (or even should've) + just defined \overline {F} to be + よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y) 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. + + This is the perspective taken in the short proof. + + However, unlike the thin case, + we crucially use the information that U is a functor + so the natural transformation squares are well-defined. +316#262unstable-262.xmlMedia202455202461indexThe Second Pass of the Portable C Compiler961Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He317#263unstable-263.xmlLinks202455202461indexPlaceholder Link553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E"An adjunction core consists of" + + functions U : A .ob → X .ob and F : X .ob → A .ob + + a family of isomorphisms (π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}374#264unstable-264.xmlNotes202455202461index318Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202461559Proposition#290unstable-290.xml20245112024511Johnson He - The following are some basic (alt. elementary) questions, whose "elevator - pitch" answers ought to roll off the tongue: - - What is a(n elementary) topos? - - - 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? - - - What is this "Kripke-Joyal semantics"? - Is this the same thing as the so-called "forcing semantics"? - - - What is this "Mitchell-Bénabou language"? - Is it the same thing as "topos logic"? - - Following Johnson Hehejohns-000E + An adjunction core extends to an adjuction iff ∀a,b ∈ A. ∀x ∈ X. + Elementary Categories, Elementary Toposes, we sketch elementary answers to these. - - Some more difficult questions we do not attempt to answer at the moment: - - What is a Grothendieck topos, and what's the difference v. - elementary toposes? - - - What is a "geometric logic" and a "geometric morphism"? - And what's the difference v. a "logical morphism"? - - - In what sense is "Kripke-Joyal semantics" a "forcing semantics"? - (What is forcing??) - - - And everything else… - - Many object to the name "higher-order intuitionistic logic". - I commutes, and iff ∀x,y ∈ X. a ∈ A. Topoi and Computation 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." - maclarty-1992 says "TODO". -328Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461II + commutes. + The adjunction is unique when it exists. + + The Johnson HeThis is a fundamental notion. - 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 notes on - realizability, - according to Loris it's relevant to Markov Decision Processes, and we - develop the view that it's a logician's view of fibrations. -original theorem is stated in terms of + \mathcal {V}-enriched categories, but we only treat the simple case + of Homset-categories here. + 326Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456555subb-000Esubb-000E.xmlI20245112024511Johnson He - Here is Blackburn et all's account: - - The concept first presented is that of Johnson He#290 + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A( a , b )×A(F x , a ) \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] & X(U a , U b )×X( x , U a ) \ar [d, "∘"] \\ + A(F x , b ) \ar [r, "π_{ x , b }"] & X( x , U b ) + \end {tikzcd} + +322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456557subbb-000Esubbb-000E.xmlII20245112024511Johnson He - "The basic case": - - A relation Z (to not be confused with the accessibility relations R for - each Kripke frame) between two Kripke frames M and M' is a *bisimulation* iff - w Z w' ⟹ (w and w' are elementarily equivalent) - - - (forth) w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v' - - (backwards) w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'Johnson He#290 + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + X( y , U a )×X( x , y ) \ar [r, "π_{ y , a }⁻¹× \overline {F} "] \ar [d, "∘"] & A(F y , a )×A(F x , F y ) \ar [d, "∘"] \\ + X( x , U a ) \ar [r, "π_{x, a}⁻¹"] & A(F x , a ) + \end {tikzcd} + +320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456561Proof#291unstable-291.xml20245112024511Johnson He - "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" +addr="hejohns">Johnson Hehejohns-000E + We use the same notation and definitions of η, ϵ, \overline {F} , \overline {U} as the + explicit proof of the precursor lemma. + The definition of \overline {F} , \overline {U} in the short proof + presupposes that U is a functor, so we're scrupulous to avoid + circularity. + + This proof is rather low tech and is lifted straight from the + paper. It's also how you'd attack the + precursor head-on. The utility is that it holds in + enriched categories. + + The forward directions are trivial. + Suppose (F, U, (π_{a, x})_{a ∈ A, x ∈ X}) extends to \overline {F} ⊣_π \overline {U}. + Then I expresses that ∀g : A(a, b). ∀f : A(F(x), a). π_{x, b}(g∘f) = \overline {U} (g)∘π_{a, x}(f). This is exactly the condition + that ∀g : A(a, b). π is natural in a at g, which is certainly + true of an adjunction. + II expresses something similar (dual). + + The nontrivial directions are the backward directions. + + (I) wts I ⟹ \overline {U} is a functor. + Suppose ∀a,b ∈ A. ∀x ∈ X. I holds. + Then in particular, ∀c ∈ A. I holds at x = U(c). + + + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A( a , b )×A(F Uc , a ) \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] & X(U a , U b )×X( Uc , U a ) \ar [d, "∘"] \\ + A(F Uc , b ) \ar [r, "π_{ Uc , b }"] & X( Uc , U b ) + \end {tikzcd} + -. - A simulation is where we drop the backwards condition. - + As does this trivial square: + + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\ + A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b) + \end {tikzcd} + + + which when pasted together, tells us magically: + + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(a, b)×A(c, a) \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\ + A(c, b) \ar [r, " \overline {U} "] & X(Uc, Ub) + \end {tikzcd} + + + Combined with the earlier proof that \overline {U} (1ₐ) = 1_{ \overline {U} (a)} (which just follows by definition), + we've shown that \overline {U} defines a functorial action on morphisms. + Actually, we saw earlier that I expresses that + π is natural in a, so we've met the preconditions of the + precursor lemma, thus there is a unique + extension \overline {F} st \overline {F} ⊣_π \overline {U}. + + + There's two ways ways to see the uniqueness situation: + + + (An intermediate notion) \overline {U} is the unique extension of + U to a functor st π is natural in a. + We've essentially seen this already, if not literally. + For any functor U', naturality in a requires the following square to hold for all g : A(a, b): + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\ + X(U'a, U'a) \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] & X(U'a, U'b) + \end {tikzcd} + + Thus, ∀g. U'(g) = \overline {U} (g). + Then, with \overline {U} a functor and π natural in a, the + precursor gives us a unique \overline {F} st \overline {F} ⊣ \overline {U} uniquely. + + + (Going backwards) Since the definitions of \overline {F} and + \overline {U} (in the precursor) are not dependent + on the other functor, the total adjunction \overline {F} ⊣ \overline {U} is + the unique one extending the core by applying the + uniqueness result of the precusor to each + functor in turn, just knowing the other is a well-defined + functor. + + + (II) wts II ⟹ \overline {F} is a functor. + This is dual. + 324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456563Proof corollarysub-000Esub-000E.xml20245112024511Johnson He - Clearly the truth of ∧,∨,◇-formulas are preserved under simulations. - Actually it's shown that the class of formulas preserved under simulations are exactly these formulas. -372Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510Johnson Hehejohns-000E + (For V-enriched categories, ) If V is thin, all diagrams commute, + thus an adjunction core always determines an adjunction. + Set isn't thin, but 2 is, thus for posets, an adjunction core + gives rise to a thin adjunction, as remarked Johnson Hehere. + I don't know how useful this lemma is for ordinary Homset categories. + 336Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511963Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -2808,7 +2992,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -2825,7 +3009,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -2844,7 +3028,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -2865,54 +3049,14 @@ addr="hejohns-0006">above lemma is in some sense the more 356Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He - Mentioned in passing as 2.1. Proposition., as "well-known". -338Proposition#267unstable-267.xml2024572024511Johnson Hehejohns-0007 - Suppose U : A → X is a functor, F : X .ob → A .ob a function, - and (πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X} a family of natural - isomorphisms. Then there is a unique extension \overline {F} of F such - that (πₓ)_{x ∈ X} is natural in x, witnessing \overline {F} ⊣ U. - - - Or dually, suppose that F is a functor, but U is just a - function on objects, and (πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}. - Then U uniquely extends to \overline {U}. - - Let's be pedantic and clarify that A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ - have the same action on morphisms as Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U. - This is conventionally left implicit, but we remark to clarify what we mean - by the family of natural isomorphisms being natural. - - 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. -344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511965Proofsub-0007sub-0007.xmlPrecursor to The Core of Adjoint Functors › the short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -2926,7 +3070,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007\usepackage {t 354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511967Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -3005,7 +3149,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -3185,155 +3329,1963 @@ addr="hejohns">Johnson Hehejohns-0 href="sub-0007.xml" type="local" addr="sub-0007">the short proof. - - However, unlike 969Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510the thin case, - we crucially use the information that U is a functor - so the natural transformation squares are well-defined. -Johnson He370Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511531Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson HeJohnson He + This is mentioned in passing, in the intro to The Core of Adjoint Functors. +358Definition#276unstable-276.xml20245112024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-000EJohnson Hehejohns-0006 + Suppose P and Q are posets, f : P → Q is a priori just a function, and u : P ← Q monotone, + such that f(x) ≤ a ⟺ x ≤ u(a). + Then f monotone as well. + + Or dually, suppose that f monotone, but u is a priori just a function. + Then u monotone as well. + 527Proofsub-0006sub-0006.xml2024572024511"An adjunction core consists of" - - functions U : A .ob → X .ob and F : X .ob → A .ob - - a family of isomorphisms (π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}Johnson Hehejohns-0006 + Suppose p₁ ≤ p₂. + Notice that p₂ ≤ u∘f(p₂), by the unit of the (thin) adjunction. + Then p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂) by the Hom bijection of the (thin) adjunction. + + (dual) + Suppose q₁ ≤ q₂. + Notice that f∘u(q₁) ≤ q₂ by the counit of the (thin) adjunction. + Then f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂) by the Hom bijection of the (thin) adjunction. + 364Proposition#277unstable-277.xml20245112024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-000E - An adjunction core extends to an adjuction iff ∀a,b ∈ A. ∀x ∈ X. - Johnson Hehejohns-0006 + Notice that I commutes, and iff ∀x,y ∈ X. a ∈ A. the above proof does not use the information that u is monotone. + Thus II - commutes. - The adjunction is unique when it exists. - +addr="hejohns-0006">the above lemma holds where f and u are just a priori functions. The original theorem is stated in terms of - \mathcal {V}-enriched categories, but we only treat the simple case - of Homset-categories here. - above lemma is in some sense the more + immediately natural question, but can be easily strengthened to + the form of this corollary. + 360subb-000Esubb-000E.xmlI20245112024511551Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He#277 - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - A( a , b )×A(F x , a ) \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] & X(U a , U b )×X( x , U a ) \ar [d, "∘"] \\ - A(F x , b ) \ar [r, "π_{ x , b }"] & X( x , U b ) - \end {tikzcd} - -Johnson He + Mentioned in passing as 2.1. Proposition., as "well-known". +362subbb-000Esubbb-000E.xmlII20245112024511533Proposition#280unstable-280.xml2024572024511Johnson He#277 - \usepackage {tikz-cd} \usepackage {unicode-math} - \begin {tikzcd} - X( y , U a )×X( x , y ) \ar [r, "π_{ y , a }⁻¹× \overline {F} "] \ar [d, "∘"] & A(F y , a )×A(F x , F y ) \ar [d, "∘"] \\ - X( x , U a ) \ar [r, "π_{x, a}⁻¹"] & A(F x , a ) - \end {tikzcd} - -Johnson Hehejohns-0007 + Suppose U : A → X is a functor, F : X .ob → A .ob a function, + and (πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X} a family of natural + isomorphisms. Then there is a unique extension \overline {F} of F such + that (πₓ)_{x ∈ X} is natural in x, witnessing \overline {F} ⊣ U. + + + Or dually, suppose that F is a functor, but U is just a + function on objects, and (πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}. + Then U uniquely extends to \overline {U}. + + Let's be pedantic and clarify that A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ + have the same action on morphisms as Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U. + This is conventionally left implicit, but we remark to clarify what we mean + by the family of natural isomorphisms being natural. + + 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. +366Proof#278unstable-278.xml20245112024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-000E - We use the same notation and definitions of η, ϵ, \overline {F} , \overline {U} as the - explicit proof of the precursor lemma. - The definition of \overline {F} , \overline {U} in the short proof - presupposes that U is a functor, so we're scrupulous to avoid - circularity. - - This proof is rather low tech and is lifted straight from Johnson Hehejohns-0007 + Suppose such a \overline {F} exists. + Then naturality of π requires that + ∀f : X(x, y). this square in Setᴬ commutes: + 535subsub-0007subsub-0007.xml2024572024511the - paper. It's also how you'd attack Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + Then clearly ∃! \overline {F} .∀f : X(x, y). \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y). + + + + (dual) Suppose such a \overline {U} exists. + Then naturality of π requires that + ∀u : A(a, b). this square in Set^{Xᵒᵖ} commutes: + 537subsubb-0007subsubb-0007.xml2024572024511the - precursor head-on. The utility is that it holds in - enriched categories. - - The forward directions are trivial. - Suppose (F, U, (π_{a, x})_{a ∈ A, x ∈ X}) extends to \overline {F} ⊣_π \overline {U}. - Then Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(-), a) \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\ + X(-, U(a)) \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] & X(-, U(b)) + \end {tikzcd} + + and clearly ∃! \overline {U} .∀u : A(a, b). \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹). + 549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511I expresses that ∀g : A(a, b). ∀f : A(F(x), a). π_{x, b}(g∘f) = \overline {U} (g)∘π_{a, x}(f). This is exactly the condition - that ∀g : A(a, b). π is natural in a at g, which is certainly - true of an adjunction. - Johnson He + We take inspiration from II expresses something similar (dual). - - The nontrivial directions are the backward directions. - - (I) wts I ⟹ \overline {U} is a functor. - Suppose ∀a,b ∈ A. ∀x ∈ X. I holds. +addr="hejohns-0006">the thin case. + We wts the F-action on morphisms is fully determined already (∃!). + That is, we have two goals: + + (∃) Show that there is a suitable F-action. + + + (!) That any other F-action is equal to it. + + As usual, let η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x)) denote the + component of the unit of the (a posteriori) adjunction + and ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)}) denote the component of the counit. + + (∃) + + Now suppose ∃f : X(x, y). + Since η_y : X(y, U∘F(y)), we have η_y∘f : X(x, U∘F(y)). + By the Hom bijection, πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y)). + This is our candidate for the F-action. + \overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f). + + + As a picture, each f : X(x, y) gives rise to the following + square in Set: + 541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\ + X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"] + \end {tikzcd} + + And our candidate \overline {F} (f) is the one we know to exist by + chasing 1_{F(y)} around the square. + + + We could directly try to show that \overline {F} defined as such is + indeed functorial--and I have tried--, but it's rather + difficult without the clarifying perspective of the Yoneda + lemma (no, seriously, although I'm sure one could do it). + + + It's instructive to try directly, if one hasn't. + The problem is that \overline {F} (f) as defined is not obviously the + composition of morphisms in any diagram-- how do you reason + about the functoriality of πₓ(F(y))⁻¹? + Instead, we want to invoke a representation theorem and work + with \overline {F} (f) as an explicit composition in some diagram. + The solution is to follow the Yoneda principle and + pass from reasoning in Set to Setᴬ. + + + Notice that the above square is + moreover natural in the variable F(y). + That is, we've actually described the F(y) component of the + natural transformation πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y), -). + As a picture, each f : X(x, y) gives + rise to the following square in Setᴬ: + 543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\ + X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + + We first make the trivial observation that A(F(x), -) and + A(F(y), -) are both Yoneda embeddings (we use the "version" + where よ : Aᵒᵖ → Setᴬ, ie the Yoneda embedding on Aᵒᵖ), + よ(F(x)) and よ(F(y)) respectively. + + + Then the Yoneda lemma tells us that by specifying an + element \overline {F} (f) : A(F(x), F(y)), this fully + determines (ie freely extends to) a natural + transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) + + Equivalently, the Yoneda principle tells us that an + element \overline {F} (f) : A(F(x), F(y)) fully failthfully + embeds as a natural transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) + + With this in mind, we notice that by definition, + よ( \overline {F} (f)) is exactly the lid to the above Setᴬ + square. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + That is, both natural transformations map 1_{F(y)} to the + same thing-- \overline {F} {F}(f)-- by definition, thus must be equal + by the Yoneda Lemma. + (In hindsight, this line of reasoning is quite + circumlocutious and you might as well start by saying such a + \overline {F} 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 \overline {F} (f) as an explicit + composition, we have enough to tools consider + functoriality. + + + Wts \overline {F} actually defines a functorial action on morphisms for a functor extending F. + Indeed, \overline {F} (1ₓ : X(x, x)) = πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)} by definition. + And suppose ∃g : X(y, z). + Then \overline {F} (g∘f : X(x, z)) viewed as a natural transformation gives rise to the following superimposed squares: + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] & 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"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left] + \end {tikzcd} + + A diagram chase verifies that the diagram indeed commutes, and + that よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g)). + By the ("going home" direction of the) Yoneda principle, \overline {F} (g∘f) = \overline {F} (f)∘ \overline {F} (g). + Thus at last, \overline {F} defines a functorial action on X-morphisms. + + + We should also check that \overline {F} is actually adjoint to U-- + ie, that (πₓ)_{x ∈ X} is indeed natural in the variable + x as well as a. + By great good fortune, by defining \overline {F} st + ∀x,y ∈ X. ∀f ∈ X(x, y). + 545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + it is indeed the case. + + + (!) + + To finish off the result, we wts for any other F' extending + F to morphisms st F' ⊣ U, witnessed by π : Xᵒᵖ → Setᴬ, + that F' = \overline {F}. + This simply follows from the observation that ∀f. naturality + of π is exactly this square, and forces + よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f)) + So this result is really a corollary of the Yoneda lemma. + The nlab page is here. +547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D + Of course in hindsight, we could've (or even should've) + just defined \overline {F} to be + よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y) 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. + + This is the perspective taken in the short proof. + + However, unlike the thin case, + we crucially use the information that U is a functor + so the natural transformation squares are well-defined. +565Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E"An adjunction core consists of" + + functions U : A .ob → X .ob and F : X .ob → A .ob + + a family of isomorphisms (π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E + An adjunction core extends to an adjuction iff ∀a,b ∈ A. ∀x ∈ X. + I commutes, and iff ∀x,y ∈ X. a ∈ A. II + commutes. + The adjunction is unique when it exists. + + The original theorem is stated in terms of + \mathcal {V}-enriched categories, but we only treat the simple case + of Homset-categories here. + 555subb-000Esubb-000E.xmlI20245112024511Johnson He#290 + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A( a , b )×A(F x , a ) \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] & X(U a , U b )×X( x , U a ) \ar [d, "∘"] \\ + A(F x , b ) \ar [r, "π_{ x , b }"] & X( x , U b ) + \end {tikzcd} + +557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#290 + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + X( y , U a )×X( x , y ) \ar [r, "π_{ y , a }⁻¹× \overline {F} "] \ar [d, "∘"] & A(F y , a )×A(F x , F y ) \ar [d, "∘"] \\ + X( x , U a ) \ar [r, "π_{x, a}⁻¹"] & A(F x , a ) + \end {tikzcd} + +561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E + We use the same notation and definitions of η, ϵ, \overline {F} , \overline {U} as the + explicit proof of the precursor lemma. + The definition of \overline {F} , \overline {U} in the short proof + presupposes that U is a functor, so we're scrupulous to avoid + circularity. + + This proof is rather low tech and is lifted straight from the + paper. It's also how you'd attack the + precursor head-on. The utility is that it holds in + enriched categories. + + The forward directions are trivial. + Suppose (F, U, (π_{a, x})_{a ∈ A, x ∈ X}) extends to \overline {F} ⊣_π \overline {U}. + Then I expresses that ∀g : A(a, b). ∀f : A(F(x), a). π_{x, b}(g∘f) = \overline {U} (g)∘π_{a, x}(f). This is exactly the condition + that ∀g : A(a, b). π is natural in a at g, which is certainly + true of an adjunction. + II expresses something similar (dual). + + The nontrivial directions are the backward directions. + + (I) wts I ⟹ \overline {U} is a functor. + Suppose ∀a,b ∈ A. ∀x ∈ X. I holds. + Then in particular, ∀c ∈ A. I holds at x = U(c). + + + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A( a , b )×A(F Uc , a ) \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] & X(U a , U b )×X( Uc , U a ) \ar [d, "∘"] \\ + A(F Uc , b ) \ar [r, "π_{ Uc , b }"] & X( Uc , U b ) + \end {tikzcd} + + + + As does this trivial square: + + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\ + A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b) + \end {tikzcd} + + + which when pasted together, tells us magically: + + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(a, b)×A(c, a) \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\ + A(c, b) \ar [r, " \overline {U} "] & X(Uc, Ub) + \end {tikzcd} + + + Combined with the earlier proof that \overline {U} (1ₐ) = 1_{ \overline {U} (a)} (which just follows by definition), + we've shown that \overline {U} defines a functorial action on morphisms. + Actually, we saw earlier that I expresses that + π is natural in a, so we've met the preconditions of the + precursor lemma, thus there is a unique + extension \overline {F} st \overline {F} ⊣_π \overline {U}. + + + There's two ways ways to see the uniqueness situation: + + + (An intermediate notion) \overline {U} is the unique extension of + U to a functor st π is natural in a. + We've essentially seen this already, if not literally. + For any functor U', naturality in a requires the following square to hold for all g : A(a, b): + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\ + X(U'a, U'a) \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] & X(U'a, U'b) + \end {tikzcd} + + Thus, ∀g. U'(g) = \overline {U} (g). + Then, with \overline {U} a functor and π natural in a, the + precursor gives us a unique \overline {F} st \overline {F} ⊣ \overline {U} uniquely. + + + (Going backwards) Since the definitions of \overline {F} and + \overline {U} (in the precursor) are not dependent + on the other functor, the total adjunction \overline {F} ⊣ \overline {U} is + the unique one extending the core by applying the + uniqueness result of the precusor to each + functor in turn, just knowing the other is a well-defined + functor. + + + (II) wts II ⟹ \overline {F} is a functor. + This is dual. + 563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E + (For V-enriched categories, ) If V is thin, all diagrams commute, + thus an adjunction core always determines an adjunction. + Set isn't thin, but 2 is, thus for posets, an adjunction core + gives rise to a thin adjunction, as remarked here. + I don't know how useful this lemma is for ordinary Homset categories. + 971Referencejacobs-1999jacobs-1999.xmlCategorical Logic and Type Theory202456202456Johnson He973Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He + "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" + +975Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He + "The basic case": + + A relation Z (to not be confused with the accessibility relations R for + each Kripke frame) between two Kripke frames M and M' is a *bisimulation* iff + w Z w' ⟹ (w and w' are elementarily equivalent) + + + (forth) w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v' + + (backwards) w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He + "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" + +977Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He + Clearly the truth of ∧,∨,◇-formulas are preserved under simulations. + Actually it's shown that the class of formulas preserved under simulations are exactly these formulas. +979Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He + Here is Blackburn et all's account: + + The concept first presented is that of 517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He + "The basic case": + + A relation Z (to not be confused with the accessibility relations R for + each Kripke frame) between two Kripke frames M and M' is a *bisimulation* iff + w Z w' ⟹ (w and w' are elementarily equivalent) + + + (forth) w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v' + + (backwards) w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He + "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" + +. + A simulation is where we drop the backwards condition. +519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He + Clearly the truth of ∧,∨,◇-formulas are preserved under simulations. + Actually it's shown that the class of formulas preserved under simulations are exactly these formulas. +884indexindex.xmlJohnson He202455202461false + + 509Personhejohnshejohns.xmlAbout Me202455202459My screen name is hejohns. This is my forest. + email: hejohns@umich.edu +TODO: I've copied over what's on https://hejohns.github.io/, but this isn't a good selection.510#263unstable-263.xmlCode202455202461indexGradescope-Utils, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly + + misc cmdline utils. colorless is nice for long git status output (it's really color-less). + 511#264unstable-264.xmlMedia202455202461indexThe Second Pass of the Portable C Compiler512#265unstable-265.xmlLinks202455202461indexPlaceholder Link569#266unstable-266.xmlNotes202455202461index513Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J523Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461Johnson HeThis is a fundamental notion. + 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 notes on + realizability, + according to Loris it's relevant to Markov Decision Processes, and we + develop the view that it's a logician's view of fibrations. +521Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He + Here is Blackburn et all's account: + + The concept first presented is that of 517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He + "The basic case": + + A relation Z (to not be confused with the accessibility relations R for + each Kripke frame) between two Kripke frames M and M' is a *bisimulation* iff + w Z w' ⟹ (w and w' are elementarily equivalent) + + + (forth) w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v' + + (backwards) w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He + "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames" + +. + A simulation is where we drop the backwards condition. +519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He + Clearly the truth of ∧,∨,◇-formulas are preserved under simulations. + Actually it's shown that the class of formulas preserved under simulations are exactly these formulas. +567Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510Johnson He531Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He + This is mentioned in passing, in the intro to The Core of Adjoint Functors. +525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 + Suppose P and Q are posets, f : P → Q is a priori just a function, and u : P ← Q monotone, + such that f(x) ≤ a ⟺ x ≤ u(a). + Then f monotone as well. + + Or dually, suppose that f monotone, but u is a priori just a function. + Then u monotone as well. + 527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 + Suppose p₁ ≤ p₂. + Notice that p₂ ≤ u∘f(p₂), by the unit of the (thin) adjunction. + Then p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂) by the Hom bijection of the (thin) adjunction. + + (dual) + Suppose q₁ ≤ q₂. + Notice that f∘u(q₁) ≤ q₂ by the counit of the (thin) adjunction. + Then f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂) by the Hom bijection of the (thin) adjunction. + 529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 + Notice that the above proof does not use the information that u is monotone. + Thus the above lemma holds where f and u are just a priori functions. + The above lemma is in some sense the more + immediately natural question, but can be easily strengthened to + the form of this corollary. + 551Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He + Mentioned in passing as 2.1. Proposition., as "well-known". +533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 + Suppose U : A → X is a functor, F : X .ob → A .ob a function, + and (πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X} a family of natural + isomorphisms. Then there is a unique extension \overline {F} of F such + that (πₓ)_{x ∈ X} is natural in x, witnessing \overline {F} ⊣ U. + + + Or dually, suppose that F is a functor, but U is just a + function on objects, and (πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}. + Then U uniquely extends to \overline {U}. + + Let's be pedantic and clarify that A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ + have the same action on morphisms as Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U. + This is conventionally left implicit, but we remark to clarify what we mean + by the family of natural isomorphisms being natural. + + 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. +539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 + Suppose such a \overline {F} exists. + Then naturality of π requires that + ∀f : X(x, y). this square in Setᴬ commutes: + 535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + Then clearly ∃! \overline {F} .∀f : X(x, y). \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y). + + + + (dual) Suppose such a \overline {U} exists. + Then naturality of π requires that + ∀u : A(a, b). this square in Set^{Xᵒᵖ} commutes: + 537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(-), a) \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\ + X(-, U(a)) \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] & X(-, U(b)) + \end {tikzcd} + + and clearly ∃! \overline {U} .∀u : A(a, b). \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹). + 549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He + We take inspiration from the thin case. + We wts the F-action on morphisms is fully determined already (∃!). + That is, we have two goals: + + (∃) Show that there is a suitable F-action. + + + (!) That any other F-action is equal to it. + + As usual, let η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x)) denote the + component of the unit of the (a posteriori) adjunction + and ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)}) denote the component of the counit. + + (∃) + + Now suppose ∃f : X(x, y). + Since η_y : X(y, U∘F(y)), we have η_y∘f : X(x, U∘F(y)). + By the Hom bijection, πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y)). + This is our candidate for the F-action. + \overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f). + + + As a picture, each f : X(x, y) gives rise to the following + square in Set: + 541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\ + X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"] + \end {tikzcd} + + And our candidate \overline {F} (f) is the one we know to exist by + chasing 1_{F(y)} around the square. + + + We could directly try to show that \overline {F} defined as such is + indeed functorial--and I have tried--, but it's rather + difficult without the clarifying perspective of the Yoneda + lemma (no, seriously, although I'm sure one could do it). + + + It's instructive to try directly, if one hasn't. + The problem is that \overline {F} (f) as defined is not obviously the + composition of morphisms in any diagram-- how do you reason + about the functoriality of πₓ(F(y))⁻¹? + Instead, we want to invoke a representation theorem and work + with \overline {F} (f) as an explicit composition in some diagram. + The solution is to follow the Yoneda principle and + pass from reasoning in Set to Setᴬ. + + + Notice that the above square is + moreover natural in the variable F(y). + That is, we've actually described the F(y) component of the + natural transformation πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y), -). + As a picture, each f : X(x, y) gives + rise to the following square in Setᴬ: + 543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\ + X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + + We first make the trivial observation that A(F(x), -) and + A(F(y), -) are both Yoneda embeddings (we use the "version" + where よ : Aᵒᵖ → Setᴬ, ie the Yoneda embedding on Aᵒᵖ), + よ(F(x)) and よ(F(y)) respectively. + + + Then the Yoneda lemma tells us that by specifying an + element \overline {F} (f) : A(F(x), F(y)), this fully + determines (ie freely extends to) a natural + transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) + + Equivalently, the Yoneda principle tells us that an + element \overline {F} (f) : A(F(x), F(y)) fully failthfully + embeds as a natural transformation よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y)) + + With this in mind, we notice that by definition, + よ( \overline {F} (f)) is exactly the lid to the above Setᴬ + square. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + That is, both natural transformations map 1_{F(y)} to the + same thing-- \overline {F} {F}(f)-- by definition, thus must be equal + by the Yoneda Lemma. + (In hindsight, this line of reasoning is quite + circumlocutious and you might as well start by saying such a + \overline {F} 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 \overline {F} (f) as an explicit + composition, we have enough to tools consider + functoriality. + + + Wts \overline {F} actually defines a functorial action on morphisms for a functor extending F. + Indeed, \overline {F} (1ₓ : X(x, x)) = πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)} by definition. + And suppose ∃g : X(y, z). + Then \overline {F} (g∘f : X(x, z)) viewed as a natural transformation gives rise to the following superimposed squares: + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] & 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"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left] + \end {tikzcd} + + A diagram chase verifies that the diagram indeed commutes, and + that よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g)). + By the ("going home" direction of the) Yoneda principle, \overline {F} (g∘f) = \overline {F} (f)∘ \overline {F} (g). + Thus at last, \overline {F} defines a functorial action on X-morphisms. + + + We should also check that \overline {F} is actually adjoint to U-- + ie, that (πₓ)_{x ∈ X} is indeed natural in the variable + x as well as a. + By great good fortune, by defining \overline {F} st + ∀x,y ∈ X. ∀f ∈ X(x, y). + 545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007\usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] \\ + X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"] + \end {tikzcd} + + it is indeed the case. + + + (!) + + To finish off the result, we wts for any other F' extending + F to morphisms st F' ⊣ U, witnessed by π : Xᵒᵖ → Setᴬ, + that F' = \overline {F}. + This simply follows from the observation that ∀f. naturality + of π is exactly this square, and forces + よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f)) + So this result is really a corollary of the Yoneda lemma. + The nlab page is here. +547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D + Of course in hindsight, we could've (or even should've) + just defined \overline {F} to be + よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y) 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. + + This is the perspective taken in the short proof. + + However, unlike the thin case, + we crucially use the information that U is a functor + so the natural transformation squares are well-defined. +565Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E"An adjunction core consists of" + + functions U : A .ob → X .ob and F : X .ob → A .ob + + a family of isomorphisms (π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E + An adjunction core extends to an adjuction iff ∀a,b ∈ A. ∀x ∈ X. + I commutes, and iff ∀x,y ∈ X. a ∈ A. II + commutes. + The adjunction is unique when it exists. + + The original theorem is stated in terms of + \mathcal {V}-enriched categories, but we only treat the simple case + of Homset-categories here. + 555subb-000Esubb-000E.xmlI20245112024511Johnson He#290 + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + A( a , b )×A(F x , a ) \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] & X(U a , U b )×X( x , U a ) \ar [d, "∘"] \\ + A(F x , b ) \ar [r, "π_{ x , b }"] & X( x , U b ) + \end {tikzcd} + +557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#290 + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + X( y , U a )×X( x , y ) \ar [r, "π_{ y , a }⁻¹× \overline {F} "] \ar [d, "∘"] & A(F y , a )×A(F x , F y ) \ar [d, "∘"] \\ + X( x , U a ) \ar [r, "π_{x, a}⁻¹"] & A(F x , a ) + \end {tikzcd} + +561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E + We use the same notation and definitions of η, ϵ, \overline {F} , \overline {U} as the + explicit proof of the precursor lemma. + The definition of \overline {F} , \overline {U} in the short proof + presupposes that U is a functor, so we're scrupulous to avoid + circularity. + + This proof is rather low tech and is lifted straight from the + paper. It's also how you'd attack the + precursor head-on. The utility is that it holds in + enriched categories. + + The forward directions are trivial. + Suppose (F, U, (π_{a, x})_{a ∈ A, x ∈ X}) extends to \overline {F} ⊣_π \overline {U}. + Then I expresses that ∀g : A(a, b). ∀f : A(F(x), a). π_{x, b}(g∘f) = \overline {U} (g)∘π_{a, x}(f). This is exactly the condition + that ∀g : A(a, b). π is natural in a at g, which is certainly + true of an adjunction. + II expresses something similar (dual). + + The nontrivial directions are the backward directions. + + (I) wts I ⟹ \overline {U} is a functor. + Suppose ∀a,b ∈ A. ∀x ∈ X. I holds. Then in particular, ∀c ∈ A. I holds at x = U(c). @@ -3429,202 +5381,708 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E + (For V-enriched categories, ) If V is thin, all diagrams commute, + thus an adjunction core always determines an adjunction. + Set isn't thin, but 2 is, thus for posets, an adjunction core + gives rise to a thin adjunction, as remarked here. + I don't know how useful this lemma is for ordinary Homset categories. + 605#267unstable-267.xmlReferences202455202461index570Referencemclarty-1990amclarty-1990a.xmlBook Review (1990) of John Bell's Toposes and Local Set Theories (1988)202461202463mclarty-1990a.pdf571Referenceawodey-2010awodey-2010.xmlCategory Theory202462202463572Referencemakkai-reyes-1977makkai-reyes-1977.xmlFirst Order Categorical Logic202462202463 + I've heard/been told this is the book for, well, the subject in the + title, and Makkai completeness. +573Referencelawvere-1970lawvere-1970.xmlQuantifiers and Sheaves202461202463295Abstract#262unstable-262.xml202461202463lawvere-1970 + 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 S 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 external 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 S (BVM/S) 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". + + 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 (∈) left behind by the + process of accumulating (∪) the power set (\mathcal {P}) 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 1 only. + + I have not read this, and have no idea what "contradictions" he's referring to. + lawvere-1970.pdf574Referencemaclane-moerdijk-1994maclane-moerdijk-1994.xmlSheaves in Geometry and Logic: A First Introduction to Topos Theory202462202463575Referencejohnstone-2002johnstone-2002.xmlSketches of an Elephant: a topos theory compendium202462202463576Referencemclarty-1990bmclarty-1990b.xmlThe Uses and Abuses of the History of Topos Theory202461202463292Abstract#260unstable-260.xml202461202463mclarty-1990b + 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. + mclarty-1990b.pdf577Referencemarquis-reyes-2012marquis-reyes-2012.xmlThe history of categorical logic: 1963-1977202461202463 + I have not read this. This might be a good source, aside from Ronnie Chen, + where the viewpoint of "generalized algebraic semantics" is developed. +marquis-reyes-2012.pdf578Referencegoldblatt-2006goldblatt-2006.xmlTopoi: The Categorial Analysis of Logic202462202463 + "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." +579Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461https://arxiv.org/pdf/1610.01213. + The nice introduction informed me of Equality Between Functionals and Equality Between Functionals Revisited. + Referenced on stack overflow. +580Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461581Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461485Abstract#282unstable-282.xml2024513202461friedman-1975 + The λ-calculus, in both its typed and untyped forms, has primarily been + regarded as an attempt to formalize the concept of rule or + process, 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. + + There is, however, another aspect of the typed 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 simple if it is given by + some closed term in the typed x-calculus. + + Several definitions of convertibility 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 + identity between rules, or processes. The relation ⊢ s = t 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." + + Square brackets denote author's footnotes. + friedman-1975.pdf582Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461487Abstract#256unstable-256.xml2024513202461statman-1985 + In this note we shall try to place Friedman's remarkable little paper + (3) 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 lays the foundation for the general + model theory of the typed λ-calculus. + + 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 + + completness theorems, + + + the solvability of higher type functional equations, and + + + logical relations. + statman-1985.pdf583Referencecarette-kiselyov-shan-2007carette-kiselyov-shan-2007.xmlFinally Tagless, Partially Evaluated2024524202461480Abstract#284unstable-284.xml2024524202461carette-kiselyov-shan-2007 + 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. + + 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. + + 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. + 481Notes#285unstable-285.xml20245242024612024525202461carette-kiselyov-shan-2007 + This is the APLAS version (conference paper). There is a roughly twice + as long journal version. + + 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 finally + tagless mean?". + + 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 Int type in the meta language as + the semantics of the DSL/embedded/object language's numeric type. + + Most people's knee jerk reaction is to encode the object language as an + ADT. This is fine and well, and can be called initial in the sense + that it is the free syntax generated by the encoded rules/initial + algebra/inductive dataype. + But the naïve TODO + 584Referencemillies-zeilberger-2015millies-zeilberger-2015.xmlFunctors are Type Refinement Systems2024511202461Johnson Hehejohns-000E - (For V-enriched categories, ) If V is thin, all diagrams commute, - thus an adjunction core always determines an adjunction. - Set isn't thin, but 2 is, thus for posets, an adjunction core - gives rise to a thin adjunction, as remarked here. - I don't know how useful this lemma is for ordinary Homset categories. - Johnson He490Abstract#258unstable-258.xml2024511202461millies-zeilberger-2015 + 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. + 401#265unstable-265.xmlReferences202455202461index491#259unstable-259.xmlSections 1-22024511202461Johnson Hemillies-zeilberger-2015What is a logical relation? + A LR is just a model of the syntax, used to prove ∀M statements. + Thus, it's usually just 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 just a clever choice of model, + to let us say something interesting/talk about what we care about. + + 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 two 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. + + This way, we can show that two different semantics and implementations + for a intset 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. + + 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? + + 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. + + 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? + + Maybe I'm finally close to being able to understand what a Kripke + logical relation is. + 375Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461https://arxiv.org/pdf/1610.01213. - The nice introduction informed me of Equality Between Functionals and 586Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461476Abstract#254unstable-254.xml2024524202461sterling-2022 + Logical relations are the main tool for proving positive + 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 synthetic Tait + computability, logical relations are manipulated as if they were sets, + making the essence of many complex logical relations arguments + accessible to non-specialists. + 477#255unstable-255.xml20245242024612024525202461sterling-2022 + I can't quite follow the phase distinction and only have a very vague + idea of what's going on starting at p. 9. + + 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 object-space/meta-space discussion is clear + and I just don't have some context, or it's actually presented almost + mystically. + The mention of Equality Between Functionals Revisited. - Referenced on stack overflow. -finally tagless encodings + and a justification for hoas doesn't seem super + on-the-point, but if it's mentioned it must be for good reason… + + Again, quotes are air-quotes, not actual quotes. + sterling-2022.pdf376Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461587Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461502Abstract#286unstable-286.xml202461202461blass-2003 + 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. + + This is currently a "would like to read, but no urgency" paper, because + Blass + Herbrand's theorem. + blass-2003.pdf377Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461588Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461288Abstract#269unstable-269.xml2024513202461friedman-1975 - The λ-calculus, in both its typed and untyped forms, has primarily been - regarded as an attempt to formalize the concept of rule or - process, 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. - - There is, however, another aspect of the typed 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 simple if it is given by - some closed term in the typed x-calculus. - - Several definitions of convertibility 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 - identity between rules, or processes. The relation ⊢ s = t 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." - - Square brackets denote author's footnotes. - friedman-1975.pdf299Abstract#287unstable-287.xml202461202461blass-1988 + 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. + 300Notes#288unstable-288.xml202461202461202461202461blass-1988 + 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. + blass-1988.pdf378Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461589Referencevon-neumann-1925von-neumann-1925.xmlAn Axiomatization of Set Theory20245282024529 + One of the early NBG papers. Maybe "the" NBG paper. + My notes on this are currently in Variables Explained Away. + + English translation in From Frege To Gödel. +290Abstract#256unstable-256.xml2024513202461statman-1985 - In this note we shall try to place Friedman's remarkable little paper - (3) 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 lays the foundation for the general - model theory of the typed λ-calculus. - - 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 - - completness theorems, - - - the solvability of higher type functional equations, and - - - logical relations. - statman-1985.pdf590Referencevan-heijenoort-1967van-heijenoort-1967.xmlFrom Frege To Gödel20245282024529 + A+. I should get a second copy to gift whenever needed. +379Referencecarette-kiselyov-shan-2007carette-kiselyov-shan-2007.xmlFinally Tagless, Partially Evaluated2024524202461593Referencequine-1960quine-1960.xmlVariables Explained Away20245282024529Johnson He + "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." + + (Yes, that 50 should be a 60(b).) + Here is the followup I wrote: +283Abstract#271unstable-271.xml2024524202461carette-kiselyov-shan-2007 - 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">591Email#257unstable-257.xml20245282024529Johnson Hequine-1960 + Thanks for sharing this nice little note, which sketches a combinatory + (classical) first-order logic. - 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: - 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. - + 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. + + 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 class existence theorem. + + (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.) + + 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. + + 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. + + 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, + + Johnson He + quine-1960.pdf + An Axiomatization of Set Theory596Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 + The first point: +284Notes#272unstable-272.xml20245242024612024525202461carette-kiselyov-shan-2007 - This is the APLAS version (conference paper). There is a roughly twice - as long journal version. - - 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 finally - tagless mean?". - - 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 Int type in the meta language as - the semantics of the DSL/embedded/object language's numeric type. - - Most people's knee jerk reaction is to encode the object language as an - ADT. This is fine and well, and can be called initial in the sense - that it is the free syntax generated by the encoded rules/initial - algebra/inductive dataype. - But the naïve TODO +root="false">595Abstract#253unstable-253.xml2024572024522street-2012 + 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. 380Referencemillies-zeilberger-2015millies-zeilberger-2015.xmlFunctors are Type Refinement Systems2024511202461Johnson He598Referencemaddy-2019maddy-2019.xmlWhat do we want a foundation to do?20245212024522293Abstract#258unstable-258.xml2024511202461millies-zeilberger-2015 - 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. - 597Abstract#261unstable-261.xml20245212024522maddy-2019 + 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. + + A nice article that can be briefly summarized as + + There is no one clear set of desiderata for a foundational + system, whatever that means + + + Classical set theory is perfectly fine as a foundational system in + the traditionally (ie ZFC) conceived sense + + + There is promise for the new 21ˢᵗ century desiderata of proof + checking (recalling Voevodsky's concerns with correctness "at the + bleeding edge"). + + 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". + + 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. + + Maddy only mentions Coq, but being 2019 it would've been nice is somewhat + redundant to mention other proof assistants eg Cubical Agda. + + I've used quotes here as air quotes, not actual quotes of Maddy. + maddy-2019.pdf601Referencedummett-91dummett-91.xmlThe Logical Basis of Metaphysics20245122024513Johnson He294#259unstable-259.xmlSections 1-22024511202461599#283unstable-283.xmlStability20245122024513Johnson Hemillies-zeilberger-2015What is a logical relation? - A LR is just a model of the syntax, used to prove ∀M statements. - Thus, it's usually just 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 just a clever choice of model, - to let us say something interesting/talk about what we care about. - - 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 two 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. - - This way, we can show that two different semantics and implementations - for a intset 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. - - 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? - - 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. - - 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 Hedummett-91 + The opening sentence is what zeilberger quotes on his work on + polarity. - 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. 382Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461603Referencejacobs-1999jacobs-1999.xmlCategorical Logic and Type Theory202456202456Johnson He279Abstract#254unstable-254.xml2024524202461sterling-2022 - Logical relations are the main tool for proving positive - 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 synthetic Tait - computability, logical relations are manipulated as if they were sets, - making the essence of many complex logical relations arguments - accessible to non-specialists. - 625#268unstable-268.xmlInbox202455202461index280#255unstable-255.xml20245242024612024525202461sterling-2022 - I can't quite follow the phase distinction and only have a very vague - idea of what's going on starting at p. 9. - - 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 object-space/meta-space discussion is clear - and I just don't have some context, or it's actually presented almost - mystically. - The mention of finally tagless encodings - and a justification for hoas doesn't seem super - on-the-point, but if it's mentioned it must be for good reason… - - Again, quotes are air-quotes, not actual quotes. - sterling-2022.pdf606Referencemakkai-reyes-1977makkai-reyes-1977.xmlFirst Order Categorical Logic202462202463 + I've heard/been told this is the book for, well, the subject in the + title, and Makkai completeness. +383Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461607Referencelawvere-1970lawvere-1970.xmlQuantifiers and Sheaves202461202463305Abstract#273unstable-273.xml202461202461blass-2003 - 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">295Abstract#262unstable-262.xml202461202463lawvere-1970 + 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 S 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 external 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 S (BVM/S) 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". + + 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 (∈) left behind by the + process of accumulating (∪) the power set (\mathcal {P}) 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 1 only. - 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. blass-2003.pdflawvere-1970.pdf608Referencemarquis-reyes-2012marquis-reyes-2012.xmlThe history of categorical logic: 1963-1977202461202463 + I have not read this. This might be a good source, aside from Ronnie Chen, + where the viewpoint of "generalized algebraic semantics" is developed. +marquis-reyes-2012.pdf384Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461609Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He308Abstract#274unstable-274.xml202461202461blass-1988 - 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">330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. 309Notes#275unstable-275.xml202461202461202461202461blass-1988 - 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. - blass-1988.pdf332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C385Referencevon-neumann-1925von-neumann-1925.xmlAn Axiomatization of Set Theory20245282024529 - One of the early NBG papers. Maybe "the" NBG paper. - My notes on this are currently in 302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Variables Explained Away. +addr="hejohns">Johnson HeTopoi and Computation 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.) - English translation in The history of categorical logic: 1963-1977, 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). + + If From Frege To Gödel. +addr="lawvere-1970">Quantifiers and Sheaves 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. 386Referencevan-heijenoort-1967van-heijenoort-1967.xmlFrom Frege To Gödel20245282024529 - A+. I should get a second copy to gift whenever needed. -321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +389Referencequine-1960quine-1960.xmlVariables Explained Away20245282024529304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson He - "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." - - (Yes, that 50 should be a 60(b).) - Here is the followup I wrote: -Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 387Email#257unstable-257.xml20245282024529312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hequine-1960 - Thanks for sharing this nice little note, which sketches a combinatory - (classical) first-order logic. - - I was surprised by the mention of von Neumann 1925 in this snippet: - - "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) - - 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. - - 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 class existence theorem. - - (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.) - - 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. - - 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. - - 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 Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." - Johnson He - quine-1960.pdf - An Axiomatization of Set Theory392Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 - The first point: -318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G391Abstract#253unstable-253.xml2024572024522street-2012 - 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. - 314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω394Referencemaddy-2019maddy-2019.xmlWhat do we want a foundation to do?20245212024522316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +393Abstract#260unstable-260.xml20245212024522maddy-2019 - 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. - - A nice article that can be briefly summarized as - - There is no one clear set of desiderata for a foundational - system, whatever that means - - - Classical set theory is perfectly fine as a foundational system in - the traditionally (ie ZFC) conceived sense - - - There is promise for the new 21ˢᵗ century desiderata of proof - checking (recalling Voevodsky's concerns with correctness "at the - bleeding edge"). - - 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". - - 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">320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. - 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 true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". - I've used quotes here as air quotes, not actual quotes of Maddy. - maddy-2019.pdfElementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +397Referencedummett-91dummett-91.xmlThe Logical Basis of Metaphysics20245122024513323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson HeJohnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +395#270unstable-270.xmlStability20245122024513Johnson Hedummett-91 - The opening sentence is what zeilberger quotes on his work on - polarity. - - 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. - 327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463399Referencejacobs-1999jacobs-1999.xmlCategorical Logic and Type Theory202456202456Johnson He325#269unstable-269.xmlSyntax202463202463hejohns-000J417#266unstable-266.xmlInbox202455202461index326#270unstable-270.xmlSemantics202463202463hejohns-000J402Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461611Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. +612Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461https://arxiv.org/pdf/1610.01213. The nice introduction informed me of 403Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461613Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461288Abstract#269unstable-269.xml2024513202461friedman-1975 +root="false">485Abstract#282unstable-282.xml2024513202461friedman-1975 The λ-calculus, in both its typed and untyped forms, has primarily been regarded as an attempt to formalize the concept of rule or process, 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">404Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461614Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461290Abstract#256unstable-256.xml2024513202461statman-1985 +root="false">487Abstract#256unstable-256.xml2024513202461statman-1985 In this note we shall try to place Friedman's remarkable little paper (405hejohns-000Ahejohns-000A.xmlFiberwise Structure202458202461615hejohns-000Ahejohns-000A.xmlFiberwise Structure202458202461Johnson He @@ -4213,7 +6893,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">407hejohns-000Bhejohns-000B.xmlLogical Frameworks as a meta-language2024527202461617hejohns-000Bhejohns-000B.xmlLogical Frameworks as a meta-language2024527202461Johnson He @@ -4240,14 +6920,14 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">409Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461619Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461279Abstract#254unstable-254.xml2024524202461sterling-2022 +root="false">476Abstract#254unstable-254.xml2024524202461sterling-2022 Logical relations are the main tool for proving positive 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">280#255unstable-255.xml20245242024612024525202461sterling-2022 +root="false">477#255unstable-255.xml20245242024612024525202461sterling-2022 I can't quite follow the phase distinction and only have a very vague idea of what's going on starting at p. 9. @@ -4299,14 +6979,14 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">410Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461620Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461305Abstract#273unstable-273.xml202461202461blass-2003 +root="false">502Abstract#286unstable-286.xml202461202461blass-2003 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">411#259unstable-259.xml621#259unstable-259.xmlFunctors are Type Refinement Systems › Sections 1-22024511202461413Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202461Johnson He - The following are some basic (alt. elementary) questions, whose "elevator - pitch" answers ought to roll off the tongue: - - What is a(n elementary) topos? - - - 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? - - - What is this "Kripke-Joyal semantics"? - Is this the same thing as the so-called "forcing semantics"? - - - What is this "Mitchell-Bénabou language"? - Is it the same thing as "topos logic"? - - Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. - - Some more difficult questions we do not attempt to answer at the moment: - - What is a Grothendieck topos, and what's the difference v. - elementary toposes? - - - What is a "geometric logic" and a "geometric morphism"? - And what's the difference v. a "logical morphism"? - - - In what sense is "Kripke-Joyal semantics" a "forcing semantics"? - (What is forcing??) - - - And everything else… - - Many object to the name "higher-order intuitionistic logic". - Topoi and Computation 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." - maclarty-1992 says "TODO". -415hejohns-0009hejohns-0009.xmlFibrations v. functors202458202461623hejohns-0009hejohns-0009.xmlFibrations v. functors202458202461Johnson He 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">442indexindex.xmlJohnson He202455202461653indexindex.xmlJohnson He202455202461false 314Personhejohnshejohns.xmlAbout Me202455202459My screen name is hejohns. This is my forest. +root="false">509Personhejohnshejohns.xmlAbout Me202455202459My screen name is hejohns. This is my forest. email: hejohns@umich.edu TODO: I've copied over what's on 315#261unstable-261.xmlCode202455202461index510#263unstable-263.xmlCode202455202461indexGradescope-Utils, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly @@ -43,7 +43,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">316#262unstable-262.xmlMedia202455202461index511#264unstable-264.xmlMedia202455202461indexThe Second Pass of the Portable C Compiler317#263unstable-263.xmlLinks202455202461index512#265unstable-265.xmlLinks202455202461indexPlaceholder Link374#264unstable-264.xmlNotes202455202461index569#266unstable-266.xmlNotes202455202461index318Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202461513Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He - The following are some basic (alt. elementary) questions, whose "elevator - pitch" answers ought to roll off the tongue: - - What is a(n elementary) topos? - - - 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? - - - What is this "Kripke-Joyal semantics"? - Is this the same thing as the so-called "forcing semantics"? - - - What is this "Mitchell-Bénabou language"? - Is it the same thing as "topos logic"? - - Following Johnson He330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) - Some more difficult questions we do not attempt to answer at the moment: - - What is a Grothendieck topos, and what's the difference v. - elementary toposes? - - - What is a "geometric logic" and a "geometric morphism"? - And what's the difference v. a "logical morphism"? - - - In what sense is "Kripke-Joyal semantics" a "forcing semantics"? - (What is forcing??) - - - And everything else… - - Many object to the name "higher-order intuitionistic logic". - The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation says "Strictly speaking, the phrase 'higher-order +addr="blass-1988">Topoi and Computation 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." - maclarty-1992 says "TODO". + underlying logic." (p. 2). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. 327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J328Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461523Expositionhejohns-0005hejohns-0005.xmlSimulations202446202461Johnson HeThis is a fundamental notion. @@ -149,7 +550,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">326Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456521Definitionhejohns-0001hejohns-0001.xmlSimulation202446202456Johnson He @@ -162,7 +563,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="false" -root="false">322Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456517Definitionhejohns-0002hejohns-0002.xmlBisimulation202446202456Johnson He @@ -182,7 +583,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">320Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456515Moral corollaryhejohns-0003hejohns-0003.xmlBisimilar Kripke frames are elementarily equivalent202446202456Johnson He @@ -197,7 +598,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">324Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456519Corollaryhejohns-0004hejohns-0004.xmlPreservation under simulations202446202456Johnson He @@ -210,7 +611,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">372Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510567Expositionhejohns-0008hejohns-0008.xmlThe core of adjoint functors2024582024510Johnson He336Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511531Lemmahejohns-0006hejohns-0006.xmlThe core of adjoint monotone functions2024572024511Johnson He @@ -235,7 +636,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">330Proposition#268unstable-268.xml2024572024511525Proposition#281unstable-281.xml2024572024511Johnson Hehejohns-0006 @@ -252,7 +653,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">332Proofsub-0006sub-0006.xml2024572024511527Proofsub-0006sub-0006.xml2024572024511Johnson Hehejohns-0006 @@ -271,7 +672,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">334Proof corollarysubb-0006subb-0006.xml2024572024511529Proof corollarysubb-0006subb-0006.xml2024572024511Johnson Hehejohns-0006 @@ -296,7 +697,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">356Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511551Lemmahejohns-0007hejohns-0007.xmlPrecursor to The Core of Adjoint Functors2024572024511Johnson He @@ -311,7 +712,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">338Proposition#267unstable-267.xml2024572024511533Proposition#280unstable-280.xml2024572024511Johnson Hehejohns-0007 @@ -339,7 +740,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">344Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511539Proofsub-0007sub-0007.xmlThe short proof (sketch)2024572024511Johnson Hehejohns-0007 @@ -353,7 +754,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">340subsub-0007subsub-0007.xml2024572024511535subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007342subsubb-0007subsubb-0007.xml2024572024511537subsubb-0007subsubb-0007.xml2024572024511Johnson Hesub-0007354Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511549Proofhejohns-000Dhejohns-000D.xmlThe verbose proof20245102024511Johnson He @@ -432,7 +833,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">346subb-000Dsubb-000D.xml20245102024511541subb-000Dsubb-000D.xml20245102024511Johnson Hehejohns-000D348sub-000Dsub-000D.xml20245102024511543sub-000Dsub-000D.xml20245102024511Johnson Hehejohns-000D350subsub-0007subsub-0007.xml2024572024511545subsub-0007subsub-0007.xml2024572024511Johnson Hesub-0007352Moralsubbb-000Dsubbb-000D.xml20245102024511547Moralsubbb-000Dsubbb-000D.xml20245102024511Johnson Hehejohns-000D @@ -626,7 +1027,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">370Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511565Lemmahejohns-000Ehejohns-000E.xmlAdjunction core20245112024511Johnson He358Definition#276unstable-276.xml20245112024511553Definition#289unstable-289.xml20245112024511Johnson Hehejohns-000E364Proposition#277unstable-277.xml20245112024511559Proposition#290unstable-290.xml20245112024511Johnson Hehejohns-000E @@ -681,10 +1082,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">360subb-000Esubb-000E.xmlI20245112024511555subb-000Esubb-000E.xmlI20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -699,10 +1100,10 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">362subbb-000Esubbb-000E.xmlII20245112024511557subbb-000Esubbb-000E.xmlII20245112024511Johnson He#277 +addr="hejohns">Johnson He#290 \usepackage {tikz-cd} \usepackage {unicode-math} \begin {tikzcd} @@ -717,7 +1118,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">366Proof#278unstable-278.xml20245112024511561Proof#291unstable-291.xml20245112024511Johnson Hehejohns-000E @@ -856,7 +1257,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">368Proof corollarysub-000Esub-000E.xml20245112024511563Proof corollarysub-000Esub-000E.xml20245112024511Johnson Hehejohns-000E @@ -875,60 +1276,214 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">401#265unstable-265.xmlReferences202455202461index605#267unstable-267.xmlReferences202455202461index375Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461https://arxiv.org/pdf/1610.01213. - The nice introduction informed me of Equality Between Functionals and Equality Between Functionals Revisited. - Referenced on stack overflow. -570Referencemclarty-1990amclarty-1990a.xmlBook Review (1990) of John Bell's Toposes and Local Set Theories (1988)202461202463mclarty-1990a.pdf571Referenceawodey-2010awodey-2010.xmlCategory Theory202462202463376Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461572Referencemakkai-reyes-1977makkai-reyes-1977.xmlFirst Order Categorical Logic202462202463 + I've heard/been told this is the book for, well, the subject in the + title, and Makkai completeness. +377Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461573Referencelawvere-1970lawvere-1970.xmlQuantifiers and Sheaves202461202463288Abstract#269unstable-269.xml2024513202461friedman-1975 - The λ-calculus, in both its typed and untyped forms, has primarily been - regarded as an attempt to formalize the concept of rule or - process, 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">295Abstract#262unstable-262.xml202461202463lawvere-1970 + 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 S 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 external 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 S (BVM/S) 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". - There is, however, another aspect of the typed 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 simple 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 (∈) left behind by the + process of accumulating (∪) the power set (\mathcal {P}) 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 1 only. + + I have not read this, and have no idea what "contradictions" he's referring to. + lawvere-1970.pdf574Referencemaclane-moerdijk-1994maclane-moerdijk-1994.xmlSheaves in Geometry and Logic: A First Introduction to Topos Theory202462202463575Referencejohnstone-2002johnstone-2002.xmlSketches of an Elephant: a topos theory compendium202462202463576Referencemclarty-1990bmclarty-1990b.xmlThe Uses and Abuses of the History of Topos Theory202461202463292Abstract#260unstable-260.xml202461202463mclarty-1990b + 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. + mclarty-1990b.pdf577Referencemarquis-reyes-2012marquis-reyes-2012.xmlThe history of categorical logic: 1963-1977202461202463 + I have not read this. This might be a good source, aside from Ronnie Chen, + where the viewpoint of "generalized algebraic semantics" is developed. +marquis-reyes-2012.pdf578Referencegoldblatt-2006goldblatt-2006.xmlTopoi: The Categorial Analysis of Logic202462202463 + "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." +579Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461https://arxiv.org/pdf/1610.01213. + The nice introduction informed me of Equality Between Functionals and Equality Between Functionals Revisited. + Referenced on stack overflow. +580Referencemclarty-1992mclarty-1992.xmlElementary Categories, Elementary Toposes2024531202461581Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461485Abstract#282unstable-282.xml2024513202461friedman-1975 + The λ-calculus, in both its typed and untyped forms, has primarily been + regarded as an attempt to formalize the concept of rule or + process, 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. + + There is, however, another aspect of the typed 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 simple if it is given by + some closed term in the typed x-calculus. Several definitions of convertibility 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">378Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461582Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461290Abstract#256unstable-256.xml2024513202461statman-1985 +root="false">487Abstract#256unstable-256.xml2024513202461statman-1985 In this note we shall try to place Friedman's remarkable little paper (379Referencecarette-kiselyov-shan-2007carette-kiselyov-shan-2007.xmlFinally Tagless, Partially Evaluated2024524202461583Referencecarette-kiselyov-shan-2007carette-kiselyov-shan-2007.xmlFinally Tagless, Partially Evaluated2024524202461283Abstract#271unstable-271.xml2024524202461carette-kiselyov-shan-2007 +root="false">480Abstract#284unstable-284.xml2024524202461carette-kiselyov-shan-2007 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">284Notes#272unstable-272.xml20245242024612024525202461carette-kiselyov-shan-2007 +root="false">481Notes#285unstable-285.xml20245242024612024525202461carette-kiselyov-shan-2007 This is the APLAS version (conference paper). There is a roughly twice as long journal version. @@ -1059,7 +1614,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">380Referencemillies-zeilberger-2015millies-zeilberger-2015.xmlFunctors are Type Refinement Systems2024511202461584Referencemillies-zeilberger-2015millies-zeilberger-2015.xmlFunctors are Type Refinement Systems2024511202461Johnson He293Abstract#258unstable-258.xml2024511202461millies-zeilberger-2015 +root="false">490Abstract#258unstable-258.xml2024511202461millies-zeilberger-2015 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">294#259unstable-259.xmlSections 1-22024511202461491#259unstable-259.xmlSections 1-22024511202461Johnson Hemillies-zeilberger-2015What is a logical relation? @@ -1155,14 +1710,14 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">382Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461586Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461279Abstract#254unstable-254.xml2024524202461sterling-2022 +root="false">476Abstract#254unstable-254.xml2024524202461sterling-2022 Logical relations are the main tool for proving positive 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">280#255unstable-255.xml20245242024612024525202461sterling-2022 +root="false">477#255unstable-255.xml20245242024612024525202461sterling-2022 I can't quite follow the phase distinction and only have a very vague idea of what's going on starting at p. 9. @@ -1214,14 +1769,14 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">383Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461587Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461305Abstract#273unstable-273.xml202461202461blass-2003 +root="false">502Abstract#286unstable-286.xml202461202461blass-2003 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">384Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461588Referenceblass-1988blass-1988.xmlTopoi and Computation202461202461308Abstract#274unstable-274.xml202461202461blass-1988 +root="false">299Abstract#287unstable-287.xml202461202461blass-1988 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">309Notes#275unstable-275.xml202461202461202461202461blass-1988 +root="false">300Notes#288unstable-288.xml202461202461202461202461blass-1988 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">385Referencevon-neumann-1925von-neumann-1925.xmlAn Axiomatization of Set Theory20245282024529 +root="false">589Referencevon-neumann-1925von-neumann-1925.xmlAn Axiomatization of Set Theory20245282024529 One of the early NBG papers. Maybe "the" NBG paper. My notes on this are currently in 386Referencevan-heijenoort-1967van-heijenoort-1967.xmlFrom Frege To Gödel20245282024529 +root="false">590Referencevan-heijenoort-1967van-heijenoort-1967.xmlFrom Frege To Gödel20245282024529 A+. I should get a second copy to gift whenever needed. 389Referencequine-1960quine-1960.xmlVariables Explained Away20245282024529593Referencequine-1960quine-1960.xmlVariables Explained Away20245282024529Johnson He @@ -1320,7 +1875,7 @@ numbered="true" show-heading="true" show-metadata="false" expanded="true" -root="false">387Email#257unstable-257.xml20245282024529591Email#257unstable-257.xml20245282024529Johnson Hequine-1960 @@ -1386,7 +1941,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">392Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 +root="false">596Referencestreet-2012street-2012.xmlThe Core of Adjoint Functors2024572024522 The first point: 391Abstract#253unstable-253.xml2024572024522street-2012 +root="false">595Abstract#253unstable-253.xml2024572024522street-2012 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">394Referencemaddy-2019maddy-2019.xmlWhat do we want a foundation to do?20245212024522598Referencemaddy-2019maddy-2019.xmlWhat do we want a foundation to do?20245212024522393Abstract#260unstable-260.xml20245212024522maddy-2019 +root="false">597Abstract#261unstable-261.xml20245212024522maddy-2019 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">397Referencedummett-91dummett-91.xmlThe Logical Basis of Metaphysics20245122024513601Referencedummett-91dummett-91.xmlThe Logical Basis of Metaphysics20245122024513Johnson He395#270unstable-270.xmlStability20245122024513599#283unstable-283.xmlStability20245122024513Johnson Hedummett-91 @@ -1505,7 +2060,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">399Referencejacobs-1999jacobs-1999.xmlCategorical Logic and Type Theory202456202456603Referencejacobs-1999jacobs-1999.xmlCategorical Logic and Type Theory202456202456Johnson He417#266unstable-266.xmlInbox202455202461index625#268unstable-268.xmlInbox202455202461index606Referencemakkai-reyes-1977makkai-reyes-1977.xmlFirst Order Categorical Logic202462202463 + I've heard/been told this is the book for, well, the subject in the + title, and Makkai completeness. +607Referencelawvere-1970lawvere-1970.xmlQuantifiers and Sheaves202461202463295Abstract#262unstable-262.xml202461202463lawvere-1970 + 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 S 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 external 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 S (BVM/S) 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". + + 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 (∈) left behind by the + process of accumulating (∪) the power set (\mathcal {P}) 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 1 only. + + I have not read this, and have no idea what "contradictions" he's referring to. + lawvere-1970.pdf608Referencemarquis-reyes-2012marquis-reyes-2012.xmlThe history of categorical logic: 1963-1977202461202463 + I have not read this. This might be a good source, aside from Ronnie Chen, + where the viewpoint of "generalized algebraic semantics" is developed. +marquis-reyes-2012.pdf402Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461609Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202463Johnson He330#278unstable-278.xmlElementary questions2024531202463Johnson Hehejohns-000C + Some basic questions, whose "elevator pitch" answers ought to roll off + the tongue: + + What is a(n elementary) topos? + + + 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? + + + What is this "Kripke-Joyal semantics"? + Is this the same thing as the so-called "forcing semantics"? + + + What is this "Mitchell-Bénabou language"? + Is it the same thing as "topos logic"? + + Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. + + Some more difficult questions we do not attempt to answer at the moment: + + What is a Grothendieck topos, and what's the difference v. + elementary toposes? + + + What is a "geometric logic" and a "geometric morphism"? + And what's the difference v. a "logical morphism"? + + + In what sense is "Kripke-Joyal semantics" a "forcing semantics"? + What is forcing?? + Why does the concept of "generic" show up everywhere? + + + What's all this algebraic geometry stuff? + + + And everything else… + + 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. + 332#279unstable-279.xmlElementary answers2024531202463Johnson Hehejohns-000C302hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. +321hejohns-000Ghejohns-000G.xmlWhat is an elementary topos?202461202463Johnson He + The usual 30-second description of an elementary topos, is that "it's like + Set" (but not quite), or "it's an axiomatization of the basic properties + of Set". + Of course, these are always the kind of descriptions that only make sense + once you've already learned what a(n elementary) topos is. + We remark that The Uses and Abuses of the History of Topos Theory is precisely about this "Set"-based + perspective. + + 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. +304Definition#271unstable-271.xmlTopos à la First Order Categorical Logic202461202463Johnson Hehejohns-000G + Okay so they only define a Grothendieck topos. We'll leave the + discussion of Grothendieck topoi to another time. + 306Definition#272unstable-272.xmlTopos à la Topoi: The Categorial Analysis of Logic 4.3202461202463Johnson Hehejohns-000G + "An elementary topos is a category \mathcal {C} such that + + (1) \mathcal {C} is finitely complete, + + + (2) \mathcal {C} is finitely co-complete, + + + (3) \mathcal {C} has exponentiation, + + + (4) \mathcal {C} has a subobject classifier. + + As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by + + (1') \mathcal {C} has a terminal object and pullbacks, + + and dually (2) replaced by + + (2') \mathcal {C} has an initial object 0, and pushouts. + + 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." + + (complete, meaning every diagram has a limit, finitely + complete, 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, + this stack overflow post.) + 308Definition#273unstable-273.xmlTopos à la Sketches of an Elephant: a topos theory compendium 2.1.1202461202463Johnson Hehejohns-000G + "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:" + + "A topos is a properly cartesian closed category with a subobject + classifier." + + "… 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." + + ("We say a category \mathcal {C} is cartesian closed 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 properly cartesian closed.") + (Recall that finite products + equalizers ⟺ finite limits.) + 310Definition#274unstable-274.xmlTopos à la Category Theory 8.17202461202463Johnson Hehejohns-000G + "A topos is a category \mathcal {E} such that + + 1. \mathcal {E} has all finite limits, + + + 2. \mathcal {E} has a subobject classifier, + + + 3. \mathcal {E} has all exponentials. + + 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." + 312Definition#275unstable-275.xmlTopos à la Elementary Categories, Elementary Toposes 13.1202461202463Johnson Hehejohns-000G + "A topos is a Cartesian closed category with a sub-object + classifier." + + Note though, that McLarty "assumes a base category with finite limits" + when talking about cartesian closed categories. + 318#276unstable-276.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory202461202463Johnson Hehejohns-000G314Definitionsub-000Gsub-000G.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω316Definition#277unstable-277.xmlTopos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 161202461202463Johnson He#276 + "A topos \mathcal {E} is a category with all finite limites, equipped with + an object Ω, with a function P which assigns to each object B + of \mathcal {E} an object PB of \mathcal {E}, and, for each object A of \mathcal {E}, with two + isomorphisms, each natural in A + \begin {align*} Sub_{ \mathcal {E} } &≅ Hom_{ \mathcal {E} }(A, Ω) &(1) \\ Hom_{ \mathcal {E} }(B × A, Ω) &≅ Hom_{ \mathcal {E} }(A, PB) &(2) \\ \end {align*} + In other words, the functors Sub_{ \mathcal {E} } and Hom_{ \mathcal {E} }(B × -, Ω), the + latter for each object B of \mathcal {E}, are required to be representable. + In the first case, the representing object Ω is the subobject + classifier, as already described in §I.3, while in the second case + PB may be called the power object of B; moreover, the + function P can be extended to a functor P : \mathcal {E} ^{op} → \mathcal {E} in + exactly one way so that (2) becomes natural in B (as well as in + A), see (7) below. Equivalently, the natural isomorphism (2) states + that PB is the exponential Ω^B, 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 §I.6. However, it will be shown in the + course of this chapter that the two sets of axioms are actually + equivalent. + + Notice that the two natural isomorphisms (1) and (2) may be combined + in a single isomorphism + \begin {align*} Sub_{ \mathcal {E} }(B × A) &≅ Hom_{ \mathcal {E} }(A, PB) &(3) \\ \end {align*} + natural in A. With B = 1, this implies (1) with Ω = P1, and hence + also (2). In other words, we could define a topos as a category with + finite limits equipped with a single operation P satisfying (3)." + + We collect the definition of a subobject classifier, which appears in + every definition, and give a standard one: +320Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. + + So comparing each definition with another, they're all about par for + course. +323hejohns-000Fhejohns-000F.xml"Higher-order intuitionistic logic" v. "topos logic"202461202463Johnson He + Many authors object to the name "higher-order intuitionistic logic". +Topoi and Computation 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). + + And Elementary Categories, Elementary Toposes 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 et al. + (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). + + 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. + + Combining the two names, I think "higher-order intuitionistic topos logic" + is at least a little more descriptive. +327hejohns-000Jhejohns-000J.xmlHigher-order intuitionistic topos logic202463202463325#269unstable-269.xmlSyntax202463202463hejohns-000J326#270unstable-270.xmlSemantics202463202463hejohns-000J611Definitionhejohns-000Ihejohns-000I.xmlSubobject classifier202463202463 + For a category \mathcal {C}, a subobject classifier is an object Ω together + with a global element true : 1 → Ω such that ∀ A ∈ \mathcal {C} . ∀ s : S ↣ A. ∃! χ_s : A → Ω. + \usepackage {tikz-cd} \usepackage {unicode-math} + \begin {tikzcd} + S \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\ + A \ar [r, "χ_s"'] & Ω + \end {tikzcd} + + is a pullback square. + + Notice that true is trivially monic since every global element is + trivially monic. + Recall that "monos are stable under pullback". + + This is from Elementary Categories, Elementary Toposes. +612Referencescherer-2017scherer-2017.xmlDeciding equivalence with sums and the empty type2024513202461https://arxiv.org/pdf/1610.01213. The nice introduction informed me of 403Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461613Referencefriedman-1975friedman-1975.xmlEquality Between Functionals2024513202461288Abstract#269unstable-269.xml2024513202461friedman-1975 +root="false">485Abstract#282unstable-282.xml2024513202461friedman-1975 The λ-calculus, in both its typed and untyped forms, has primarily been regarded as an attempt to formalize the concept of rule or process, 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">404Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461614Referencestatman-1985statman-1985.xmlEquality Between Functionals Revisited2024513202461290Abstract#256unstable-256.xml2024513202461statman-1985 +root="false">487Abstract#256unstable-256.xml2024513202461statman-1985 In this note we shall try to place Friedman's remarkable little paper (405hejohns-000Ahejohns-000A.xmlFiberwise Structure202458202461615hejohns-000Ahejohns-000A.xmlFiberwise Structure202458202461Johnson He @@ -1640,7 +2769,7 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">407hejohns-000Bhejohns-000B.xmlLogical Frameworks as a meta-language2024527202461617hejohns-000Bhejohns-000B.xmlLogical Frameworks as a meta-language2024527202461Johnson He @@ -1667,14 +2796,14 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">409Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461619Referencesterling-2022sterling-2022.xmlNaïve Logical Relations in Synthetic Tait Computability2024524202461279Abstract#254unstable-254.xml2024524202461sterling-2022 +root="false">476Abstract#254unstable-254.xml2024524202461sterling-2022 Logical relations are the main tool for proving positive 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">280#255unstable-255.xml20245242024612024525202461sterling-2022 +root="false">477#255unstable-255.xml20245242024612024525202461sterling-2022 I can't quite follow the phase distinction and only have a very vague idea of what's going on starting at p. 9. @@ -1726,14 +2855,14 @@ numbered="false" show-heading="true" show-metadata="true" expanded="false" -root="false">410Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461620Referenceblass-2003blass-2003.xmlResource Consciousness in Classical Logic202461202461305Abstract#273unstable-273.xml202461202461blass-2003 +root="false">502Abstract#286unstable-286.xml202461202461blass-2003 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">411#259unstable-259.xml621#259unstable-259.xmlFunctors are Type Refinement Systems › Sections 1-22024511202461413Expositionhejohns-000Chejohns-000C.xmlElementary² toposes2024531202461Johnson He - The following are some basic (alt. elementary) questions, whose "elevator - pitch" answers ought to roll off the tongue: - - What is a(n elementary) topos? - - - 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? - - - What is this "Kripke-Joyal semantics"? - Is this the same thing as the so-called "forcing semantics"? - - - What is this "Mitchell-Bénabou language"? - Is it the same thing as "topos logic"? - - Following Elementary Categories, Elementary Toposes, we sketch elementary answers to these. - - Some more difficult questions we do not attempt to answer at the moment: - - What is a Grothendieck topos, and what's the difference v. - elementary toposes? - - - What is a "geometric logic" and a "geometric morphism"? - And what's the difference v. a "logical morphism"? - - - In what sense is "Kripke-Joyal semantics" a "forcing semantics"? - (What is forcing??) - - - And everything else… - - Many object to the name "higher-order intuitionistic logic". - Topoi and Computation 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." - maclarty-1992 says "TODO". -415hejohns-0009hejohns-0009.xmlFibrations v. functors202458202461623hejohns-0009hejohns-0009.xmlFibrations v. functors202458202461Johnson He 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">747Referencejacobs-1999jacobs-1999.xmlCategorical Logic and Type Theory202456202456982Referencejacobs-1999jacobs-1999.xmlCategorical Logic and Type Theory202456202456Johnson He \ 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 @@ + + 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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 @@ + + +343Referencelawvere-1970lawvere-1970.xmlQuantifiers and Sheaves202461202463295Abstract#262unstable-262.xml202461202463lawvere-1970 + 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 S 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 external 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 S (BVM/S) 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". + + 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 (∈) left behind by the + process of accumulating (∪) the power set (\mathcal {P}) 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 1 only. + + I have not read this, and have no idea what "contradictions" he's referring to. + lawvere-1970.pdf341hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. + \ 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 @@ + + +348Referencemaclane-moerdijk-1994maclane-moerdijk-1994.xmlSheaves in Geometry and Logic: A First Introduction to Topos Theory202462202463346Definitionsub-000Gsub-000G.xmlWhat is an elementary topos? › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48202461202463Johnson He#276 + "[A topos \mathcal {E} is a category] with the following properties + + (i) \mathcal {E} has all finite limits and colimits, + + + (ii) \mathcal {E} has exponentials, + + + (iii) \mathcal {E} has a subobject classifier 1 → Ω \ 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">503Referencemaddy-2019maddy-2019.xmlWhat do we want a foundation to do?20245212024522712Referencemaddy-2019maddy-2019.xmlWhat do we want a foundation to do?20245212024522393Abstract#260unstable-260.xml20245212024522maddy-2019 +root="false">597Abstract#261unstable-261.xml20245212024522maddy-2019 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 @@ + + +340Referencemakkai-reyes-1977makkai-reyes-1977.xmlFirst Order Categorical Logic202462202463 + I've heard/been told this is the book for, well, the subject in the + title, and Makkai completeness. + \ 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 index 0000000000000000000000000000000000000000..5a8824fbfb4a9eeae0c71417f17520b8dc8e52d7 GIT binary patch literal 702485 zcma%?Ly#x{k8azxZQHhO+qP}nwr$()zHQsK`TlwJu$axPlFB|+Ij6pmDu{^DGSV?Y zk&a)-7eTQRFc8=qSwZpe(94+GnY&mJu>WVGL@#D(<6`PWKrd!v=wd2jYHV*}%Et%g z?BZlV}hsGrK#WZ^D^vj 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zt1k`6H)H)W1Huukya%mJY(3yJuxmSj<>#seZ{lF-0`HBrS%HsR6G3ndE5K@63DDf^ zc%Um<)~*9SZhe7ZS@|cx_2XqRNqlv?z{jmG5G*T;NyBq~ExENUzhEY?JWjBzAAp4u zF78dFXZ<%x)G>TOu&m6+N*e-{%(KDD=J*+MgS8m|%dK@_#g&i$p~tLVMgNe1u5OoL zSw0Z-h!3%}hmFa{X^;@Y@&;fznnK{SI?8Cm21I5KqJP{xhjYZmco3ikBdDJP_z=!z z%3%4hF-ySGYY9{`5f&GiNL@zwSUCyR!^PVPmiDI5pX_T0@Wff00r+4*tLp(D7jGw6 zRv#fX01-9@7FHNPz6kj;&q1&}J}jT}^n-|D+CaFI)hVzhIo?Ikma_@*yqMG+K6=BA znpS{ieNF^HVEm}-0UuTlm@;^^tR5s-Zp;D8@;LB0;AgBau>Aups~52B!t*XfC%4`R z7lM^x1k36GEKDHi!8{w*NI8B+);}jp0L!j-BUBGVk;A9w|gkE3C};5-gW< zf^Z_&XV?N9KSRTFdKIwjS}Hst+UBR?dps}SE;3!XwG+6e>{_Z3s~@bt3Vfgz(+1^7 zurviM=aawz=4b_c>{z110;dDWd1l*yhr{t2@bPQ@Ixa7byd%~&qY!`td?s#}9-kmNaiiH*jK>%*co4I8?;wFw^ + +356Referencemarquis-reyes-2012marquis-reyes-2012.xmlThe history of categorical logic: 1963-1977202461202463 + I have not read this. This might be a good source, aside from Ronnie Chen, + where the viewpoint of "generalized algebraic semantics" is developed. +marquis-reyes-2012.pdf355Personronnie-chenronnie-chen.xmlRonnie Chen202461202463353hejohns-000Hhejohns-000H.xmlWhy is it called an "elementary" topos?202461202463Johnson HeTopoi and Computation 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.) + + This is corroborated by The history of categorical logic: 1963-1977, 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). + + If Quantifiers and Sheaves 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. + \ No newline at end of file diff --git a/_site/forest/mclarty-1990a.pdf b/_site/forest/mclarty-1990a.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3f005afc089217caf58447a42d1cdf65f1f49199 GIT binary patch literal 1602526 zcmcG!c{JPI_xRhYk`TmHsjO;VTm!A#7XnR z7B;_qvip}siDWI2tbdQf|Mxq~-;&LLBrE&h(!T$`vyw&yFhBg;qV2!GGC%y=gzbL@BkAq8^go^1{g!P1W#acZ%|m^o6Jn45+i7gF zg^i_T9{$hk|JzjjHvfAntnICB|JMxt?&v==WbPfCav~#MV&cE9V3RUp(m$FDognwdwl6=P^Mf#uqhn4o#=_yys57_55 z9=`6j4A}=ip`SWxBvN(;JiESaJ7p zNZ!)5ZacKyky|C4*-Sg&{=m=A;=?mD>vm(vr4zupKR=miECic$1$=f{zm_*C;-HT9 z0;1*5ywo?Q>o{Dmg;P=8{w~$RpR?l6UbjSoHL>Bwt)E+V-~dtX_ZM07VKIIFNqttw zAD;e`k@e+ZZ?>cOMYfCFSp%yY-nm~N*Rsb?o91d+I}9{MU+I#z#M2u+)7!n%XCJk2uQdxjKC4^UiN;eM>zhhnLyyG#l0DZ7XduRrN@-~&scrTvcA 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    a pure waste of space

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    related

  • (the answer is yes, you should.)
    • -

    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

    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 @@

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  • (the answer is yes, you should.)
    • -

    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

    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 @@

  • (the answer is yes, you should.)
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    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

    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 @@

    reviews

    Shoutout to kaobook.

    - -
    View PDF
    + +
    View PDF
    @@ -99,7 +99,7 @@

    reviews

  • (the answer is yes, you should.)
    • -

    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

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    site version: 2024-06-01

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    a pure waste of space

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    related

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    +

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    related

  • (the answer is yes, you should.)
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    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

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    related

  • (the answer is yes, you should.)
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    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

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    related

  • (the answer is yes, you should.)
    • -

    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

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    related

  • (the answer is yes, you should.)
    • -

    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

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    related

  • (the answer is yes, you should.)
    • -

    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

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    related

  • (the answer is yes, you should.)
    • -

    site version: 2024-06-01

    +

    site version: 2024-06-03

    a pure waste of space

    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 @@ https://hejohns.github.io/forest/blass-1988.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:50-04:00 https://hejohns.github.io/forest/blass-2003.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:50-04:00 https://hejohns.github.io/forest/carette-kiselyov-shan-2007.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:50-04:00 https://hejohns.github.io/forest/friedman-1975.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:50-04:00 https://hejohns.github.io/forest/hoffman-1999.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:50-04:00 https://hejohns.github.io/forest/lambek-1989.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:50-04:00 + + +https://hejohns.github.io/forest/lawvere-1970.pdf +2024-06-03T05:14:50-04:00 https://hejohns.github.io/forest/maddy-2019.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:50-04:00 + + +https://hejohns.github.io/forest/marquis-reyes-2012.pdf +2024-06-03T05:14:50-04:00 + + +https://hejohns.github.io/forest/mclarty-1990a.pdf +2024-06-03T05:14:50-04:00 + + +https://hejohns.github.io/forest/mclarty-1990b.pdf +2024-06-03T05:14:50-04:00 https://hejohns.github.io/forest/millies-zeilberger-2015.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:50-04:00 https://hejohns.github.io/forest/pfenning-elliott-1988.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:51-04:00 https://hejohns.github.io/forest/quine-1960.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:51-04:00 https://hejohns.github.io/forest/statman-1985.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:51-04:00 https://hejohns.github.io/forest/sterling-2022.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:51-04:00 https://hejohns.github.io/forest/wadler-1989-ocr.pdf -2024-06-01T16:41:42-04:00 +2024-06-03T05:14:51-04:00 https://hejohns.github.io/google1be2851b772a6a30.html