From 12638e4667845725378a4c70a89c1f2e95f7ba74 Mon Sep 17 00:00:00 2001 From: Steve Awodey Date: Mon, 19 Aug 2024 20:05:54 -0400 Subject: [PATCH 1/6] Update Outline.md --- GroupoidModel/Outline.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/GroupoidModel/Outline.md b/GroupoidModel/Outline.md index 80b5e21..3330866 100644 --- a/GroupoidModel/Outline.md +++ b/GroupoidModel/Outline.md @@ -42,7 +42,7 @@ We will need at least the following: -/ /- -# 3. HoTT 0 +# 3. HoTT_0 No idea how to do this in practice! But it would be nice if the user could: - interact with types, terms, and type families, rather than @@ -56,7 +56,7 @@ But it would be nice if the user could: /- # Targets -Here are some things that should be doable in HoTT 0: +Here are some things that should be doable in HoTT_0: - HoTT Book real numbers and cumulative hierarchy, - univalent set theory, - structure identity principle for general algebraic structures, From d45e9a0a1ec4b4fbad76196ed09dd496a5bdd2a9 Mon Sep 17 00:00:00 2001 From: jlh Date: Fri, 23 Aug 2024 10:25:05 -0400 Subject: [PATCH 2/6] add .gitignore for pdflatex --- blueprint/print/.gitignore | 9 +++++++++ blueprint/print/print.pdf | Bin 0 -> 6776 bytes blueprint/src/.gitignore | 1 + blueprint/src/print.tex | 6 +++--- 4 files changed, 13 insertions(+), 3 deletions(-) create mode 100644 blueprint/print/.gitignore create mode 100644 blueprint/print/print.pdf create mode 100644 blueprint/src/.gitignore diff --git a/blueprint/print/.gitignore 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b/blueprint/src/content.tex deleted file mode 100644 index b5cd27b..0000000 --- a/blueprint/src/content.tex +++ /dev/null @@ -1,7 +0,0 @@ -% In this file you should put the actual content of the blueprint. -% It will be used both by the web and the print version. -% It should *not* include the \begin{document} -% -% If you want to split the blueprint content into several files then -% the current file can be a simple sequence of \input. Otherwise It -% can start with a \section or \chapter for instance. \ No newline at end of file diff --git a/blueprint/src/groupoid_model.tex b/blueprint/src/groupoid_model.tex new file mode 100644 index 0000000..2456c49 --- /dev/null +++ b/blueprint/src/groupoid_model.tex @@ -0,0 +1,1193 @@ +In this section we construct a natural model in $\Pshgrpd$ the presheaf category +indexed by the category $\grpd$ of (small) groupoids. +We will build the classifier for display maps in the style of +Hofmann and Streicher \cite{hofmannstreicher1996} and Awodey \cite{awodey2023hofmannstreicheruniverses}. +To interpret the type constructors, +we will make use of the weak factorization system on $\grpd$ - +which comes from restricting the ``classical Quillen model structure'' on +$\cat$ \cite{joyalnlabmodelstructuresoncat} to $\grpd$. + +\medskip + +\subsection{Classifying display maps} + +\medskip + +\begin{notation*} + We will have two universe sizes - one small and one large. + We denote the category of small sets as $\set$ and the large sets as $\Set$ + (in the previous sections this would have been $\Set_{\lambda}$ and $\Set$ respectively). + We denote the category of small categories as $\cat$ and the large categories as $\Cat$. + We denote the category of small groupoids as $\grpd$. %% and the large groupoids as $\GRPD$. + + We are primarily working in the category of large presheaves indexed by small groupoids, + which we will denote by + \[ \Pshgrpd = [\grpd^{\op}, \Set]\] + + In this section, $\Term$ and $\Type$ and so on will refer to the natural model semantics in this + specific model. +\end{notation*} + +\medskip + +\begin{defn}[Pointed] + We will take the category of pointed small categories $\ptcat$ + to have objects as pairs $(\catC \in \cat, c \in \catC)$ + and morphisms as pairs + \[ (F : \catC_{1} \to \catC_{0}, \phi : F c_{1} \to c_{0}) \co (\catC_{1}, c_{1}) \to (\catC_{0}, c_{0})\] + Then the category of pointed small groupoids $\ptgrpd$ will be the full subcategory + of objects $(\Ga, c)$ with $\Ga$ a groupoid. +\end{defn} + +\medskip + +\begin{defn}[The display map classifier] + We would like to define a natural transformation in + $\Pshgrpd$ + \[ \tp \co \Term \to \Type \] + with representable fibers. + + Consider the functor that forgets the point + \[ + U \co \ptgrpd \to \grpd + \quad \quad + \text{ + in \quad $\Cat$. + } + \] + If we apply the Yoneda embedding $\yo \co \Cat \to \PshCat$ to $U$ + we obtain + \[ U \circ \co [ - , \ptgrpd] \to [ - , \grpd ] + \quad \quad + \text{ + in \quad $\PshCat$. + } + \] + Since any small groupoid is also a large category $\grpd \hookrightarrow \Cat$, + we can restrict $\Cat$ indexed presheaves to be $\grpd$ indexed presheaves. + We define $\tp \co \Term \to \Type$ as the image of $U \circ$ under this restriction. + % https://q.uiver.app/#q=WzAsNixbMCwwLCJcXENhdCJdLFsxLDAsIlxcUHNoQ2F0Il0sWzIsMCwiXFxQc2hncnBkIl0sWzAsMSwiXFxncnBkIl0sWzEsMSwiWy0sXFxncnBkXSJdLFsyLDEsIlxcVHlwZSJdLFswLDEsInkiXSxbMSwyLCJcXHRleHRzZntyZXN9Il0sWzMsNCwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFs0LDUsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0= + \[\begin{tikzcd}[row sep = tiny] + \Cat & \PshCat & \Pshgrpd \\ + \grpd & {[-,\grpd]} & \Type + \arrow["\yo", from=1-1, to=1-2] + \arrow["{\textsf{res}}", from=1-2, to=1-3] + \arrow[maps to, from=2-1, to=2-2] + \arrow[maps to, from=2-2, to=2-3] + \end{tikzcd}\] + Note that $\Term$ and $\Type$ are not representable in $\Pshgrpd$. +\end{defn} + +\medskip + +\begin{rmk} + By Yoneda we can identify maps with representable domain into the + type classifier + \[ A : \yo \Ga \to \Type \quad \quad \text{ in } \quad \Pshgrpd \] + with functors + \[ A : \Ga \to \grpd \quad \quad \text{ in } \quad \Cat \] + +\end{rmk} + +\medskip + +\begin{defn}[Grothendieck construction] + From $\catC$ a small category and $F : \catC \to \cat$ + a functor, we construct a small category $\int F$. + For any $c$ in $\catC$ we refer to $F c$ as the fiber over $c$. + The objects of $\int F$ consist of pairs $(c \in \catC, x \in F c)$, + and morphisms between $(c, x)$ and $(d, y)$ + are pairs $(f : c \to d, \phi : F \, f \, x \to y)$. + This makes the following pullback in $\Cat$ + % https://q.uiver.app/#q=WzAsMTEsWzEsMSwiXFxpbnQgRiJdLFsxLDIsIlxcY2F0QyJdLFsyLDEsIlxccHRjYXQiXSxbMiwyLCJcXGNhdCJdLFswLDEsIihjLHgpIl0sWzAsMiwiYyJdLFsxLDAsIihjLHgpIl0sWzIsMCwiKEZjLCB4KSJdLFsxLDNdLFszLDEsIihDLGMpIl0sWzMsMiwiQyJdLFswLDFdLFswLDJdLFsyLDNdLFsxLDMsIkYiLDJdLFs0LDUsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbNiw3LCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzAsMywiIiwyLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV0sWzksMTAsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0= + \[\begin{tikzcd} + & {(c,x)} & {(Fc, x)} \\ + {(c,x)} & {\int F} & \ptcat & {(C,c)} \\ + c & \catC & \cat & C \\ + & {} + \arrow[maps to, from=1-2, to=1-3] + \arrow[maps to, from=2-1, to=3-1] + \arrow[from=2-2, to=2-3] + \arrow[from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow[from=2-3, to=3-3] + \arrow[maps to, from=2-4, to=3-4] + \arrow["F"', from=3-2, to=3-3] + \end{tikzcd}\] +\end{defn} + +\medskip + +\begin{defn}[Grothendieck construction for groupoids] + Let $\Ga$ be a groupoid and $A \co \Ga \to \grpd$ a functor, + we can compose $F$ with the inclusion $i \co \grpd \hookrightarrow \Cat$ + and form the Grothendieck construction which we denote as + \[ \Ga \cdot A := \int i \circ A \quad \quad \disp{A} \co \Ga \cdot A \to \Ga\] + This is also a small groupoid since the underlying morphisms are + pairs of morphisms from groupoids $\Ga$ and $A x$ for $x \in \Ga$. + Furthermore the pullback factors through (pointed) groupoids. + % https://q.uiver.app/#q=WzAsNyxbMCwwLCJcXEdhIFxcY2RvdCBBIl0sWzAsMSwiXFxHYSJdLFsxLDAsIlxccHRncnBkIl0sWzEsMSwiXFxncnBkIl0sWzAsMl0sWzIsMCwiXFxwdENhdCJdLFsyLDEsIlxcQ2F0Il0sWzAsMV0sWzAsMl0sWzIsM10sWzEsMywiQSIsMl0sWzAsMywiIiwyLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV0sWzIsNV0sWzMsNl0sWzUsNl0sWzIsNiwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d + \[\begin{tikzcd} + {\Ga \cdot A} & \ptgrpd & \ptcat \\ + \Ga & \grpd & \cat \\ + {} + \arrow[from=1-1, to=1-2] + \arrow["\disp{A}"', from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow[from=1-2, to=1-3] + \arrow[from=1-2, to=2-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-2, to=2-3] + \arrow[from=1-3, to=2-3] + \arrow["A"', from=2-1, to=2-2] + \arrow[from=2-2, to=2-3] + \end{tikzcd}\] +\end{defn} + +\medskip + +\begin{cor}[The display map classifier is presentable] + For any small groupoid $\Ga$ and $A : \yo \Ga \to \Type$, + the pullback of $\tp$ along $A$ can be given by the representable + map $\yo \disp{A}$. + % https://q.uiver.app/#q=WzAsNSxbMCwwLCJcXHlvIFxcR2EgXFxjZG90IEEiXSxbMCwxLCJcXHlvIFxcR2EiXSxbMSwwLCJcXFRlcm0iXSxbMSwxLCJcXFR5cGUiXSxbMCwyXSxbMCwxLCJcXGRpc3AgQSIsMl0sWzAsMl0sWzIsMywiXFx0cCIsMl0sWzEsMywiQSIsMl0sWzAsMywiIiwyLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d + \[\begin{tikzcd} + {\yo \Ga \cdot A} & \Term \\ + {\yo \Ga} & \Type \\ + {} + \arrow[from=1-1, to=1-2] + \arrow["{\yo \disp A}"', from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow["\tp"', from=1-2, to=2-2] + \arrow["A"', from=2-1, to=2-2] + \end{tikzcd}\] +\end{cor} +\begin{proof} + Consider the pullback in $\Cat$ + % https://q.uiver.app/#q=WzAsNSxbMCwwLCJcXEdhIFxcY2RvdCBBIl0sWzAsMSwiXFxHYSJdLFsxLDAsIlxccHRncnBkIl0sWzEsMSwiXFxncnBkIl0sWzAsMl0sWzAsMV0sWzAsMl0sWzIsM10sWzEsMywiQSIsMl0sWzAsMywiIiwyLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d + \[\begin{tikzcd} + {\Ga \cdot A} & \ptgrpd \\ + \Ga & \grpd \\ + {} + \arrow[from=1-1, to=1-2] + \arrow[from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow[from=1-2, to=2-2] + \arrow["A"', from=2-1, to=2-2] + \end{tikzcd}\] + We send this square along $\textsf{res} \circ \yo$ in the following + % https://q.uiver.app/#q=WzAsNSxbMiwxLCJcXFBzaGdycGQiXSxbMSwyXSxbMiwwLCJcXFBzaENhdCJdLFswLDAsIlxcQ2F0Il0sWzAsMSwiXFxncnBkIl0sWzMsMiwiXFx5byJdLFsyLDAsIlxcdGV4dHNme3Jlc30iXSxbMywwXSxbNCwzXSxbNCwwLCJcXHlvIl1d + \[\begin{tikzcd} + \Cat && \PshCat \\ + \grpd && \Pshgrpd \\ + & {} + \arrow["\yo", from=1-1, to=1-3] + \arrow[from=1-1, to=2-3] + \arrow["{\textsf{res}}", from=1-3, to=2-3] + \arrow[from=2-1, to=1-1] + \arrow["\yo", from=2-1, to=2-3] + \end{tikzcd}\] + The Yoneda embedding $\yo : \Cat \to \PshCat$ preserves pullbacks, + as does $\textsf{res}$ since it is a right adjoint + (with left Kan extension $\io_{!} \dashv \textsf{res}_{\io}$). +\end{proof} + +\medskip + +\subsection{Groupoid fibrations} + +\begin{defn}[Fibration] + Let $p : \catC_{1} \to \catC_{0}$ be a functor. + We say $p$ is a \textit{split Grothendieck fibration} + if we have a dependent function $\lift{a}{f}$ satisfying the following: + for any object $a$ in $\catC_{1}$ and morphism $f : p \, a \to y$ + in the base $\catC_{0}$ + we have $\lift{a}{f} : a \to b$ in $\catC_{1}$ such that $p (\lift{a}{f}) = f$ + and moreover $\lift{a}{g \circ f} = \lift{b}{g} \circ \lift{a}{f}$ + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJhIl0sWzEsMCwiYiJdLFswLDEsIngiXSxbMSwxLCJ5Il0sWzAsMSwiXFxsaWZ0e2F9e2Z9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzAsMiwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFsyLDMsImYiLDJdLFsxLDMsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifSwiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzQsNiwiIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfSwic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9LCJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= + \[\begin{tikzcd} + a & b \\ + x & y + \arrow[""{name=0, anchor=center, inner sep=0}, "{\lift{a}{f}}", dashed, from=1-1, to=1-2] + \arrow[maps to, from=1-1, to=2-1] + \arrow[dashed, maps to, from=1-2, to=2-2] + \arrow[""{name=1, anchor=center, inner sep=0}, "f"', from=2-1, to=2-2] + \arrow[shorten <=4pt, shorten >=4pt, Rightarrow, dashed, maps to, from=0, to=1] + \end{tikzcd}\] + + In particular, we are intereseted in split Grothendieck fibrations of groupoids, + which are the same as \textit{isofibrations} + (replace all the morphisms with isomorphisms in the definition). + + Unless specified otherwise, + by a \textit{fibration} we will mean a split Grothendieck fibration of groupoids. + Let us denote the category of fibrations over a groupoid $\Ga$ as $\Fib_{\Ga}$, + which is a full subcategory of the slice $\grpd / \Ga$. + We will decorate an arrow with $\twoheadrightarrow$ + to indicate it is a fibration. +\end{defn} + +\medskip + +Note that $\disp{A} \co \Ga \cdot A \to \Ga$ is a fibration, +since for any $(x \in \Ga, a \in A \, x)$ and $f \co x \to y$ in $\Ga$ +we have a morphism $(f, \id_{A \, f \, a}) : (x, a) \to (y, A \, f \, a)$ +lifting $f$. Furthermore + +\begin{prop} + There is an adjoint equivalence + % https://q.uiver.app/#q=WzAsMixbMCwwLCJbXFxHYSwgXFxncnBkXSJdLFsyLDAsIlxcRmliX1xcR2EiXSxbMCwxLCJcXGRpc3B7fSIsMCx7Im9mZnNldCI6LTJ9XSxbMSwwLCJcXHRleHRzZntmaWJlcn0iLDAseyJvZmZzZXQiOi0yfV0sWzAsMSwiXFxzaW1lcSIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6Im5vbmUifSwiaGVhZCI6eyJuYW1lIjoibm9uZSJ9fX1dXQ== + \[\begin{tikzcd} + {[\Ga, \grpd]} && {\Fib_\Ga} + \arrow["{\disp{}}", shift left=2, from=1-1, to=1-3] + \arrow["\simeq"{description}, draw=none, from=1-1, to=1-3] + \arrow["{\fiber}", shift left=2, from=1-3, to=1-1] + \end{tikzcd}\] + where for each fibration $\de : \De \to \Ga$ and each + object $x \in \Ga$ + \[\fiber_{\de} \, x = \text{\emph {full subcategory}} \, \{a \in \De \st \de \, a = x\}\] + It follows that all fibrations are pullbacks of the classifier + $U : \ptgrpd \to \grpd$, when viewed as morphisms in $\Cat$. +\end{prop} + +\medskip + +Pullback of fibrations along groupoid functors is not strictly coherent, +in the sense that for $\tau : \Xi \to \De$ and $\si : \De \to \Ga$ +and a fibration $p \in \Fib_{\Ga}$ we only have an isomorphism +\[ \tau^{*} \si^{*} p \iso (\si \circ \tau)^{*} p\] +rather than equality. + +In order to interpret reindexing/substitution strictly, +it is convenient to work with classifiers $[\Ga,\grpd]$ +instead of fibrations. + +\medskip + +\begin{prop}[Strictly coherent pullback] + Let $\si : \De \to \Ga$ be a functor between groupoids. + Since display maps are pullbacks + of the classifier $U : \ptgrpd \to \grpd$ we have + the pasting diagram + % https://q.uiver.app/#q=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 + \[\begin{tikzcd} + {\De.A\si} & {\Ga.A} & \ptgrpd \\ + \De & \Ga & \grpd + \arrow["{\si_A}"', dashed, from=1-1, to=1-2] + \arrow[bend left, from=1-1, to=1-3] + \arrow["{\disp{A\si}}"{description}, two heads, from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow[from=1-2, to=1-3] + \arrow["{\disp{A}}"{description}, two heads, from=1-2, to=2-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-2, to=2-3] + \arrow[from=1-3, to=2-3] + \arrow["\si"', from=2-1, to=2-2] + \arrow["A"', from=2-2, to=2-3] + \end{tikzcd}\] + This gives us a functor $\circ \si : [\Ga,\grpd] \to [\De,\grpd]$ + which is our strict version of pullback. +\end{prop} + +\medskip + +\begin{cor}[Fibrations are stable under pullback] + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJbXFxHYSwgXFxncnBkXSJdLFsyLDAsIlxcRmliX1xcR2EiXSxbMCwxLCJbXFxEZSwgXFxncnBkXSJdLFsyLDEsIlxcRmliX1xcRGUiXSxbMSwwLCJcXHRleHRzZntmaWJlcn0iLDJdLFswLDIsIlxcY2lyYyBcXHNpIiwyXSxbMiwzLCJcXGRpc3B7fSIsMl0sWzEsMywiXFxzaV4qIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d + \[\begin{tikzcd} + {[\Ga, \grpd]} && {\Fib_\Ga} \\ + {[\De, \grpd]} && {\Fib_\De} + \arrow["{\circ \si}"', from=1-1, to=2-1] + \arrow["{\fiber}"', from=1-3, to=1-1] + \arrow["{\si^*}", dashed, from=1-3, to=2-3] + \arrow["{\disp{}}"', from=2-1, to=2-3] + \end{tikzcd}\] + We can deduce a corresponding fact about fibrations: + since fibrations are closed under isomorphism, + and since any pullback in $\grpd$ of a fibration $p$ is isomorphic to + the display map $\disp{\fiber{p} \circ \si}$, + any pullback of a fibration is a fibration. +\end{cor} + +\medskip + +A strict interpretation of type theory would require +$\Si$ and $\Pi$-formers to be stable under pullback (Beck-Chevalley). +Thus we again define these as operations on classifiers $[\Ga,\grpd]$. + +\medskip + +\begin{defn}[$\Si$-former operation] + Then given $A : \Ga \to \grpd$ and $B : \Ga\cdot A \to \grpd$ we define + $\Si_{A}B : \Ga \to \grpd$ such that + $\Si_{A}B$ acts on objects by forming fiberwise Grothendieck constructions + \[ \Si_{A}B (x) := A(x) \cdot B \circ x_{A} \] + where $x_{A} : A(x) \to \Ga \cdot A$ takes $f : a_{0} \to a_{1}$ to + $(\id_{x},f) : (x,a_{0}) \to (x,a_{1})$ + % https://q.uiver.app/#q=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 + \[\begin{tikzcd} + {A(x) \cdot B \circ x_A} & {\Ga.A.B} & \bullet \\ + {A(x)} & {\Ga.A} & \grpd \\ + \terminal & \Ga & \grpd + \arrow[dashed, from=1-1, to=1-2] + \arrow["{\disp{B \circ x_A}}"{description}, two heads, from=1-1, to=2-1] + \arrow[from=1-2, to=1-3] + \arrow["{\disp{B}}"{description}, from=1-2, to=2-2] + \arrow["{x_A}"', from=2-1, to=2-2] + \arrow["{!}"{description}, from=2-1, to=3-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-1, to=3-2] + \arrow["B", from=2-2, to=2-3] + \arrow["{\disp{A}}"{description}, two heads, from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["x"', from=3-1, to=3-2] + \arrow["A"', from=3-2, to=3-3] + \end{tikzcd}\] + $\Si_{A}B$ acts on morphism $f : x \to y$ in $\Ga$ and $(a \in A(x), b \in B(x,a))$ by + \[ \Si_{A}B \, f \, (a,b) := (A \, f \, a, B \, (f,\id_{{A \, f \, a}}) \, b) \] + and for morphism $(\al : a_{0} \to a_{1} \in A(x),\be : B \, (\id_{x},\al) \, b_{0} \to b_{1} \in B(x,a_{1}))$ + in $\Si_{A}B \, x$ + \[ \Si_{A}B \, f \, (\al,\be) := (A \, f \, \al, B \, (f,\id_{{A \, f \, a_{1}}}) \, \be) \] + + Let us also define the natural transformation + $\fst : \Si_{A}B \to A$ by + \[ \fst_{x} : (a, b) \mapsto a \] +\end{defn} + +\medskip + +\begin{prop}[Fibrations are closed under composition] + The corresponding fact about fibrations is that + the composition of two fibrations is a fibration. + % https://q.uiver.app/#q=WzAsMyxbMCwwLCJcXFhpIl0sWzAsMSwiXFxEZSJdLFsxLDEsIlxcR2EiXSxbMCwxLCIiLDAseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMSwyLCIiLDAseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMCwyLCIiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV1d + \[\begin{tikzcd} + \Xi \\ + \De & \Ga + \arrow[two heads, from=1-1, to=2-1] + \arrow[dashed, two heads, from=1-1, to=2-2] + \arrow[two heads, from=2-1, to=2-2] + \end{tikzcd}\] +\end{prop} + +We can compare the two fibrations +\[ \disp{B} \circ \disp{A} + \quad \quad + \text{ and } + \quad \quad + \disp{\Si_{A}(B)} +\] +An object in the composition would look like $((x,a),b)$ +for $x \in \Ga$, $a \in A(x)$ and $b \in B(x,a)$, +whereas an object in $\Ga \cdot {\Si_{A}(B)}$ would instead be $(x,(a,b))$. + +\medskip +\begin{prop}[Strict Beck-Chevalley for $\Si$] + Let $\si : \De \to \Ga$, $A : \Ga \to \grpd$ and $B : \Ga\cdot A \to \grpd$. + Then + \[ (\Si_{A}B) \circ \si = \Si_{A \circ \si}(B \circ \si_{A})\] + where $\si_{A}$ is uniquely determined by the pullback in + % https://q.uiver.app/#q=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 + \[\begin{tikzcd} + & {\De \cdot A \si \cdot B \circ \si_A} & {\Ga.A.B} \\ + & {\De \cdot A \si} & {\Ga.A} & \grpd \\ + \grpd & \De & \Ga & \grpd + \arrow["{\si_{A \cdot B}}"', from=1-2, to=1-3] + \arrow["{\disp{B \circ \si_A}}"{description}, two heads, from=1-2, to=2-2] + \arrow["{\disp{B}}"{description}, two heads, from=1-3, to=2-3] + \arrow["{\si_A}"', from=2-2, to=2-3] + \arrow["{\disp{A\si}}"{description}, from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["B", from=2-3, to=2-4] + \arrow["{\disp{A}}"{description}, two heads, from=2-3, to=3-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-3, to=3-4] + \arrow["{(\Si_A B) \circ \si}"', shift right, from=3-2, to=3-1] + \arrow["{\Si_{A \circ \si} (B \circ \si_A)}", shift left, from=3-2, to=3-1] + \arrow["\si"', from=3-2, to=3-3] + \arrow["A"', from=3-3, to=3-4] + \end{tikzcd}\] +\end{prop} +\begin{proof} + By checking pointwise at $x \in \De$, this boils down to showing + \[(\si x)_{A} = \si_{A} \circ x_{A \circ \si} : A (\si x) \to \Ga \cdot A\] + % https://q.uiver.app/#q=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 + \[\begin{tikzcd} + {A (\si x)} & {\De \cdot A \si} & {\Ga.A} & \grpd \\ + \terminal & \De & \Ga & \grpd + \arrow["{x_{A \si}}"', from=1-1, to=1-2] + \arrow["{(\si x)_A}", bend left, from=1-1, to=1-3] + \arrow["{!}"{description}, from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow["{\si_A}"', from=1-2, to=1-3] + \arrow["{\disp{A\si}}"{description}, from=1-2, to=2-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-2, to=2-3] + \arrow["B", from=1-3, to=1-4] + \arrow["{\disp{A}}"{description}, two heads, from=1-3, to=2-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-3, to=2-4] + \arrow["x"', from=2-1, to=2-2] + \arrow["\si"', from=2-2, to=2-3] + \arrow["A"', from=2-3, to=2-4] + \end{tikzcd}\] + + which holds because of the universal property of pullback. +\end{proof} + +\medskip + +\begin{defn}[$\Pi$-former operation]\label{pi_classifier_op} + Given $A : \Ga \to \grpd$ and $B : \Ga \cdot A \to \grpd$ + we will define $\Pi_{A} B : \Ga \to \grpd$ such that + for any $C : \Ga \to \grpd$ we have an isomorphism + \[ [\Ga \cdot A, \grpd](\disp{A}\circ C, B) \iso + [\Ga, \grpd](C, \Pi_{A} B)\] + natural in both $B$ and $C$. +\end{defn} +\begin{proof} + $\Pi_{A}B$ acts on objects by taking fiberwise sections + \[ \Pi_{A}B (x) := \{s \in [A (x), \Si_{A}B (x)] \st \fst_x \circ s = \id_{A (x)}\} \] + Where we have taken the full subcategory of the functor category $[A (x), \Si_{A}B (x)]$. + This is a groupoid since any natural transformation + of functors into groupoids are natural isomorphisms. + + $\Pi_{A}B$ acts on morphisms via conjugation + % https://q.uiver.app/#q=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 + \[\begin{tikzcd} + x &&& {\Pi_A B (x)} &&& {A (x)} && {\Si_{A}B (x)} \\ + & {} & {} \\ + y &&& {\Pi_A B(y)} &&& {A (y)} && {\Si_{A}B (y)} + \arrow["f", from=1-1, to=3-1] + \arrow["{\Si_AB(f) \circ - \circ A (f^{-1})}", from=1-4, to=3-4] + \arrow["s"{description}, from=1-7, to=1-9] + \arrow["{\Si_A B(f)}"{description}, from=1-9, to=3-9] + \arrow["{\Pi_A B}", maps to, from=2-2, to=2-3] + \arrow["{A(f^{-1})}"{description}, from=3-7, to=1-7] + \arrow["{\Pi_A B (f) (s)}"', dashed, from=3-7, to=3-9] + \end{tikzcd}\] + Note that conjugation is functorial and invertible. +\end{proof} + +\medskip + +\begin{cor}[Fibrations are closed under pushforward] + Stated in terms of fibrations, we have + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXFhpIl0sWzAsMSwiXFxEZSJdLFsxLDEsIlxcR2EiXSxbMSwwLCJcXEdhXyEgXFxzaV8qIFxcdGF1Il0sWzAsMSwiXFx0YXUiLDIseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMSwyLCJcXHNpIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzMsMiwiXFxzaV8qIFxcdGF1IiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV1d + \[\begin{tikzcd} + \Xi & {\Ga_! \si_* \tau} \\ + \De & \Ga + \arrow["\tau"', two heads, from=1-1, to=2-1] + \arrow["{\si_* \tau}", two heads, from=1-2, to=2-2] + \arrow["\si"', two heads, from=2-1, to=2-2] + \end{tikzcd}\] + with the universal property of pushforward + \[ \Fib_{\De}(\si^{*}\rho, \tau) \iso \Fib_{\Ga}(\rho, \si_{*} \tau)\] + natural in both $\tau$ and $\rho$. +\end{cor} + +\medskip + +\begin{prop}[Strict Beck-Chevalley for $\Pi$] + Let $\si : \De \to \Ga$, $A : \Ga \to \grpd$ and $B : \Ga\cdot A \to \grpd$. + Then + \[ (\Pi_{A}B) \circ \si = \Pi_{A \circ \si}(B \circ \si_{A})\] + where $\si_{A}$ is uniquely determined by the pullback in + % https://q.uiver.app/#q=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 + \[\begin{tikzcd} + & {\De \cdot A \si \cdot B \circ \si_A} & {\Ga.A.B} \\ + & {\De \cdot A \si} & {\Ga.A} & \grpd \\ + \grpd & \De & \Ga & \grpd + \arrow["{\si_{A \cdot B}}"', from=1-2, to=1-3] + \arrow["{\disp{B \circ \si_A}}"{description}, two heads, from=1-2, to=2-2] + \arrow["{\disp{B}}"{description}, two heads, from=1-3, to=2-3] + \arrow["{\si_A}"', from=2-2, to=2-3] + \arrow["{\disp{A\si}}"{description}, from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["B", from=2-3, to=2-4] + \arrow["{\disp{A}}"{description}, two heads, from=2-3, to=3-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-3, to=3-4] + \arrow["{\Pi_A B \circ \si}"', shift right, from=3-2, to=3-1] + \arrow["{\Pi_{A \circ \si} (B \circ \si_A)}", shift left, from=3-2, to=3-1] + \arrow["\si"', from=3-2, to=3-3] + \arrow["A"', from=3-3, to=3-4] + \end{tikzcd}\] +\end{prop} +\begin{proof} + By checking pointwise, this boils down to Beck-Chevalley for $\Si$. +\end{proof} + +\medskip + +\begin{prop}[All objects are fibrant] + Let $\terminal$ denote the terminal groupoid, + namely that with a single object and morphism. + Then the unique map $\Ga \to \terminal$ is a fibration. +\end{prop} + +\medskip + +\begin{defn}[Interval] + Let the interval groupoid $\Interval$ be the small groupoid with two objects + and a single non-identity isomorphism. + There are two distinct morphisms $\de_{0}, \de_{1} : \terminal \to \Interval$ + and a natural isomorphism $i : \de_{0} \Rightarrow \de_{1}$. + Note that $\de_{0}$ and $\de_{1}$ both form adjoint equivalences with the unique + map $! : \Interval \to \terminal$. + + Denote by $\Two$ the small groupoid with two objects and only identity morphisms. + Then let $\partial : \Two \to \Interval$ be the unique map factoring $\de_{0}$ and $\de_{1}$. + + % https://q.uiver.app/#q=WzAsNCxbMSwxLCJcXFR3byJdLFsyLDIsIlxcSW50ZXJ2YWwiXSxbMCwxLCJcXHRlcm1pbmFsIl0sWzEsMCwiXFx0ZXJtaW5hbCJdLFswLDEsIlxccGFydGlhbCIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsyLDEsIlxcZGVfMCIsMix7ImN1cnZlIjoxfV0sWzMsMSwiXFxkZV8xIiwwLHsiY3VydmUiOi0xfV0sWzIsMF0sWzMsMF1d + \[\begin{tikzcd} + & \terminal \\ + \terminal & \Two \\ + && \Interval + \arrow[from=1-2, to=2-2] + \arrow["{\de_1}", bend left, from=1-2, to=3-3] + \arrow[from=2-1, to=2-2] + \arrow["{\de_0}"', bend right, from=2-1, to=3-3] + \arrow["\partial"{description}, dashed, from=2-2, to=3-3] + \end{tikzcd}\] +\end{defn} + +\medskip + +\begin{prop}[Path object fibration] + Let $\Ga$ be a small groupoid. + Recall that $\grpd$ is Cartesian closed, + so we can take the image of the above diagram under the functor $\Ga^{-}$. + % https://q.uiver.app/#q=WzAsNCxbMSwxLCJcXEdhIFxcdGltZXMgXFxHYSJdLFsxLDIsIlxcR2EiXSxbMiwxLCJcXEdhIl0sWzAsMCwiXFxHYV5cXEludGVydmFsIl0sWzAsMV0sWzAsMl0sWzMsMSwiXFxHYV57XFxkZV8wfSIsMix7ImN1cnZlIjoxLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywyLCJcXEdhXntcXGRlXzF9IiwwLHsiY3VydmUiOi0xLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywwLCJcXEdhXlxccGFydGlhbCIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9LCJoZWFkIjp7Im5hbWUiOiJlcGkifX19XV0= + \[\begin{tikzcd} + {\Ga^{\Interval}} \\ + & {\Ga \times \Ga} & \Ga \\ + & \Ga + \arrow["{\Ga^\partial}"{description}, dashed, two heads, from=1-1, to=2-2] + \arrow["{\Ga^{\de_1}}", bend left, two heads, from=1-1, to=2-3] + \arrow["{\Ga^{\de_0}}"', bend right, two heads, from=1-1, to=3-2] + \arrow[from=2-2, to=2-3] + \arrow[from=2-2, to=3-2] + \end{tikzcd}\] + Then the indicated morphisms are fibrations, + and $\Ga^{\de_{0}}, \Ga^{\de_{1}}$ form adjoint equivalences + with $\Ga^{!} : \Ga \to \Ga^{\Interval}$. +\end{prop} + +\medskip + +\subsection{Polynomial endofunctors} + +\begin{defn}[Polynomial endofunctor on a morphism + in an locally Cartesian closed category] + Let $\catC$ be a locally Cartesian closed category + (we will take presheaves on small groupoids). + This means for each morphism $t : B \to A$ we have an adjoint triple + % https://q.uiver.app/#q=WzAsMyxbMCwyXSxbMSwyLCJcXGNhdEMgLyBBIl0sWzEsMCwiXFxjYXRDIC8gQiJdLFsyLDEsInRfKiIsMCx7Im9mZnNldCI6LTV9XSxbMiwxLCJ0XyEiLDIseyJvZmZzZXQiOjV9XSxbMSwyLCJ0XioiLDFdLFs1LDMsIiIsMix7ImxldmVsIjoxLCJzdHlsZSI6eyJuYW1lIjoiYWRqdW5jdGlvbiJ9fV0sWzQsNSwiIiwyLHsibGV2ZWwiOjEsInN0eWxlIjp7Im5hbWUiOiJhZGp1bmN0aW9uIn19XV0= + \[\begin{tikzcd} + & {\catC / B} \\ + \\ + {} & {\catC / A} + \arrow[""{name=0, anchor=center, inner sep=0}, "{t_*}", bend left, shift left=5, from=1-2, to=3-2] + \arrow[""{name=1, anchor=center, inner sep=0}, "{t_!}"', bend right, shift right=5, from=1-2, to=3-2] + \arrow[""{name=2, anchor=center, inner sep=0}, "{t^*}"{description}, from=3-2, to=1-2] + \arrow["\dashv"{anchor=center}, draw=none, from=1, to=2] + \arrow["\dashv"{anchor=center}, draw=none, from=2, to=0] + \end{tikzcd}\] + where $t^{*}$ is pullback, and $t_{!}$ is composition with $t$. + + Let $t : B \to A$ be a morphism in $\catC$. + Then define $\Poly{t} : \catC \to \catC$ be the composition + \[ + \Poly{t} := A_{!} \circ t_{*} \circ B^{*} + \quad \quad \quad + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGNhdEMiXSxbMSwwLCJcXGNhdEMgLyBCIl0sWzIsMCwiXFxjYXRDIC8gQSJdLFszLDAsIlxcY2F0QyJdLFswLDEsIkJeKiJdLFsxLDIsInRfKiJdLFsyLDMsIkFfISJdXQ== + \begin{tikzcd} + \catC & {\catC / B} & {\catC / A} & \catC + \arrow["{B^*}", from=1-1, to=1-2] + \arrow["{t_*}", from=1-2, to=1-3] + \arrow["{A_!}", from=1-3, to=1-4] + \end{tikzcd}\] +\end{defn} + +\medskip + +\begin{prop} + The data of a map into the polynomial applied to an object in $\catC$ + % https://q.uiver.app/#q=WzAsMixbMCwwLCJYIl0sWzEsMCwiXFxQb2x5e3R9IFkiXSxbMCwxXV0= + \[\begin{tikzcd} + X & {\Poly{t} Y} + \arrow["\phi", from=1-1, to=1-2] + \end{tikzcd}\] + corresponds to + % https://q.uiver.app/#q=WzAsMyxbMCwwLCJYIl0sWzIsMCwiXFxQb2x5e3R9IFkiXSxbMSwxLCJBIl0sWzAsMSwiXFxwaGkiXSxbMSwyLCJ0XyogQl4qIFkiXSxbMCwyLCJcXGFsIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d + \[\begin{tikzcd} + X && {\Poly{t} Y} \\ + & A + \arrow["\phi", from=1-1, to=1-3] + \arrow["\al"', dashed, from=1-1, to=2-2] + \arrow["{t_* B^* Y}", from=1-3, to=2-2] + \end{tikzcd}\] + Applying the adjunction $A_{!} \dashv A^{*}$, this corresponds to + % https://q.uiver.app/#q=WzAsMyxbMCwwLCJCXyEgdF4qXFxhbCJdLFsyLDAsIkIgXFx0aW1lcyBZIl0sWzEsMSwiQiJdLFswLDEsIlxcdGlsZGV7XFxwaGl9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsMiwiQl4qIFkiXSxbMCwyLCJ0XipcXGFsIiwyXV0= + \[ + \al : X \to A + \quad \quad \text{ and } + \quad \quad + \begin{tikzcd} + {B_! t^*\al} && {B \times Y} \\ + & B + \arrow["{\tilde{\phi}}", dashed, from=1-1, to=1-3] + \arrow["{t^*\al}"', from=1-1, to=2-2] + \arrow["{B^* Y}", from=1-3, to=2-2] + \end{tikzcd}\] + Applying the adjunction $t^{*} \dashv t_{*}$, + this corresponds to + % https://q.uiver.app/#q=WzAsMixbMCwwLCJCXyEgdF4qXFxhbCJdLFsyLDAsIlkiXSxbMCwxLCJcXHRpbGRle1xcdGlsZGV7XFxwaGl9fSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ== + \[ + \al : X \to A + \quad \quad \text{ and } + \quad \quad + \begin{tikzcd} + {B_! t^*\al} && Y + \arrow["{\beta}", dashed, from=1-1, to=1-3] + \end{tikzcd}\] +\end{prop} + +\medskip + +\begin{prop}[$\Poly{\tp}\Type$ classifies dependent types]\label{tp_classifies} + Specialized to $\tp : \Term \to \Type$ in $\Pshgrpd$, + the previous proposition says that a map + from a representable + $\Ga \to \Poly{\tp}\Type$ + corresponds to the data of + \[ + A : \Ga \to \Type + \quad \quad + \text{ and } + \quad \quad + B : \Ga \cdot A \to \Type + \] + which by Yoneda corresponds to the data in $\Cat$ of + \[ + A : \Ga \to \grpd + \quad \quad + \text{ and } + \quad \quad + B : \Ga \cdot A \to \grpd + \] + + Furthermore, if $\si : \De \to \Ga$ were a representable map, + then we have a naturality square +% https://q.uiver.app/#q=WzAsNixbMywwLCJcXFBvbHl7XFx0cH0gW1xcR2EsXFxjYXRDXSAiXSxbMywxLCJcXFBvbHl7XFx0cH0gW1xcRGUsXFxjYXRDXSAiXSxbMCwxLCJcXERlIl0sWzAsMCwiXFxHYSJdLFsyLDAsIlxcU2lfe0EgXFxpbiBbXFxHYSxcXGdycGRdfSAgW1xcR2EuQSxcXGNhdENdIl0sWzIsMSwiXFxTaV97QSBcXGluIFtcXERlLFxcZ3JwZF19ICBbXFxEZS5BLFxcY2F0Q10iXSxbMiwzLCJcXHNpIl0sWzAsMSwiXFxQb2x5e1xcdHB9ey0gXFxjaXJjIFxcc2l9IiwxXSxbNCwwLCJcXGlzbyJdLFs1LDEsIlxcaXNvIiwyXSxbNCw1LCIoLSBcXGNpcmMgXFxzaSwtIFxcY2lyYyBcXHNpX0EpIiwxXV0= +\[\begin{tikzcd} + \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\catC]} & {\Poly{\tp} \Type \, \Ga } \\ + \De && {\Si_{A \in [\De,\grpd]} [\De.A,\catC]} & {\Poly{\tp} \Type \, \De } + \arrow["\iso", from=1-3, to=1-4] + \arrow["{(- \circ \si,- \circ \si_A)}"{description}, from=1-3, to=2-3] + \arrow["{\Poly{\tp}{- \circ \si}}"{description}, from=1-4, to=2-4] + \arrow["\si", from=2-1, to=1-1] + \arrow["\iso"', from=2-3, to=2-4] +\end{tikzcd}\] +\end{prop} + +\medskip + +\subsection{Pi and Sigma structure} %% TODO fix subsection title + +\medskip + +\begin{lemma}\label{BC_iff_nat_trans} + Let $\catC$ be a large category, + and let $[-,\catC] \in \Pshgrpd$ be the restriction of the Yoneda embedding + $\yo : \Cat \to \PshCat$. + Let $F$ be an operation that takes a groupoid $\Ga$, + a functor $A : \Ga \to \grpd$ + and $B : \Ga \cdot A \to \catC$ and returns a functor + $F_{A} B : \Ga \to \catC$. + + Then $\tilde{F} : \Poly{\tp} [-,\catC] \to [-,\catC]$ + \[ \tilde{F}_{\Ga} (A,B) = F_{A}B \] + defines a natural transformation + if and only if $F$ satisfies the strict Beck-Chevalley condition + \[ (F_{A} B) \circ \si = F_{A \circ \si} (B \circ \si_{A})\] + where $\si_{A}$ is given by + % https://q.uiver.app/#q=WzAsNixbMSwwLCJcXEdhXFxjZG90IEEiXSxbMSwxLCJcXEdhIl0sWzIsMSwiXFxncnBkIl0sWzIsMCwiXFxwdGdycGQiXSxbMCwxLCJcXERlIl0sWzAsMCwiXFxEZSBcXGNkb3QgQVxcY2lyYyBcXHNpIl0sWzAsMSwiIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzEsMiwiQSIsMl0sWzMsMl0sWzAsM10sWzQsMSwiXFxzaSIsMl0sWzUsNCwiIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzUsMCwiXFxzaV9BIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d +\[\begin{tikzcd} + {\De \cdot A\circ \si} & {\Ga\cdot A} & \ptgrpd \\ + \De & \Ga & \grpd + \arrow["{\si_A}", dashed, from=1-1, to=1-2] + \arrow[two heads, from=1-1, to=2-1] + \arrow[from=1-2, to=1-3] + \arrow[two heads, from=1-2, to=2-2] + \arrow[from=1-3, to=2-3] + \arrow["\si"', from=2-1, to=2-2] + \arrow["A"', from=2-2, to=2-3] +\end{tikzcd}\] +\end{lemma} +\begin{proof} + Using \cref{tp_classifies} +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\catC]} & {\Poly{\tp} [\Ga,\catC] } & {[\Ga,\catC]} \\ + \De && {\Si_{A \in [\De,\grpd]} [\De.A,\catC]} & {\Poly{\tp} [\De,\catC] } & {[\De,\catC] } + \arrow[from=1-3, to=1-4] + \arrow["F"{description}, bend left, from=1-3, to=1-5] + \arrow["{(- \circ \si,- \circ \si_A)}"{description}, from=1-3, to=2-3] + \arrow["{F_\Ga}"', from=1-4, to=1-5] + \arrow["{\Poly{\tp}{- \circ \si}}"', from=1-4, to=2-4] + \arrow["{- \circ \si}", from=1-5, to=2-5] + \arrow["\si", from=2-1, to=1-1] + \arrow[from=2-3, to=2-4] + \arrow["F"{description}, bend right, from=2-3, to=2-5] + \arrow["{F_\De}", from=2-4, to=2-5] +\end{tikzcd}\] +\end{proof} + +\begin{defn}[Interpretation of $\Pi$ types] + We define the natural transformation + $\Pi : \Poly{\tp} \Type \to \Type$ + as that which is induced (\cref{BC_iff_nat_trans}) by the $\Pi$-former operation (\cref{pi_classifier_op}). + + Then we define the natural transofrmation + $\la : \Poly{\tp} \Type \to \Type$ + as the natural transformation induced by the following operation: + given $A : \Ga \to \grpd$ and $\be : \Ga \cdot A \to \ptgrpd$, + $\la_{A}\be : \Ga \to \ptgrpd$ will be the functor such that on objects + $x \in \Ga$ + \[ \la_{A}\be \, (x) := (\Pi_{A} B \, (x) , a \mapsto (a, b (x , a)))\] + where $B := U \circ \be : \Ga \cdot A \to \grpd$ + and $b (x , a)$ is the point in $\be (x , a)$. + On morphisms $f : x \to y$ in $\Ga$ we have + \[ \la_{A}\be \, (f) := (\Pi_{A} B \, (f) , \eta )\] + where $\eta : \Pi_{A} B \, f \, s_{x} \to s_{y}$ is a natural isomorphism + between functors $A_{y} \to \Si_{A} B y$ given on objects $a \in A_{y}$ by + \[ \eta_{a} := (\id_{a}, \id_{b (y , a)})\] + + These combine to give us a pullback square +% https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXFBvbHl7XFx0cH17XFxUZXJtfSJdLFswLDEsIlxcUG9seXtcXHRwfXtcXFR5cGV9Il0sWzEsMCwiXFxUZXJtIl0sWzEsMSwiXFxUeXBlIl0sWzAsMSwiXFxQb2x5e1xcdHB9e1xcdHB9IiwyXSxbMCwyLCJcXGxhIl0sWzIsMywiXFx0cCJdLFsxLDMsIlxcUGkiLDJdLFswLDMsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ== +\[\begin{tikzcd} + {\Poly{\tp}{\Term}} & \Term \\ + {\Poly{\tp}{\Type}} & \Type + \arrow["\la", from=1-1, to=1-2] + \arrow["{\Poly{\tp}{\tp}}"', from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow["\tp", from=1-2, to=2-2] + \arrow["\Pi"', from=2-1, to=2-2] +\end{tikzcd}\] +\end{defn} +\begin{proof} + We should check that the $\la$ operation satisfied Beck-Chevalley. + This follows from the $\Pi$ satisfying Beck-Chevalley and extensionality + results for functors. + + The square commutes and is a pullback if and only it pointwise + commutes and pointwise gives pullbacks, i.e. + for each groupoid $\Ga$ + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXFBvbHl7XFx0cH17XFxUZXJtfSBcXEdhIl0sWzAsMSwiXFxQb2x5e1xcdHB9e1xcVHlwZX0gXFxHYSJdLFsxLDAsIltcXEdhLFxccHRncnBkXSJdLFsxLDEsIltcXEdhLFxcZ3JwZF0iXSxbMCwxLCJcXFBvbHl7XFx0cH17XFx0cH1fXFxHYSIsMl0sWzAsMiwiXFxsYV9cXEdhIl0sWzIsMywiVSBcXGNpcmMgLSJdLFsxLDMsIlxcUGlfXFxHYSIsMl0sWzAsMywiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d +\[\begin{tikzcd} + {\Poly{\tp}{\Term} \, \Ga} & {[\Ga,\ptgrpd]} \\ + {\Poly{\tp}{\Type} \, \Ga} & {[\Ga,\grpd]} + \arrow["{\la_\Ga}", from=1-1, to=1-2] + \arrow["{\Poly{\tp}{\tp}_\Ga}"', from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow["{U \circ -}", from=1-2, to=2-2] + \arrow["{\Pi_\Ga}"', from=2-1, to=2-2] +\end{tikzcd}\] + +by \cref{tp_classifies} this holds if and only if +% https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXFNpX3tBIFxcaW4gW1xcR2EsXFxncnBkXX0gIFtcXEdhLkEsXFxwdGdycGRdIl0sWzAsMSwiXFxTaV97QSBcXGluIFtcXEdhLFxcZ3JwZF19ICBbXFxHYS5BLFxcZ3JwZF0iXSxbMSwwLCJbXFxHYSxcXHB0Z3JwZF0iXSxbMSwxLCJbXFxHYSxcXGdycGRdIl0sWzAsMSwiKC0gXFxjaXJjIFxcc2ksLSBcXGNpcmMgXFxzaV9BKSIsMl0sWzAsMiwiXFxsYSJdLFsyLDMsIlUgXFxjaXJjIC0iXSxbMSwzLCJcXFBpIiwyXSxbMCwzLCIiLDAseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0= +\[\begin{tikzcd} + {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\ptgrpd]} & {[\Ga,\ptgrpd]} \\ + {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\grpd]} & {[\Ga,\grpd]} + \arrow["\la", from=1-1, to=1-2] + \arrow["{(- \circ \si,- \circ \si_A)}"', from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow["{U \circ -}", from=1-2, to=2-2] + \arrow["\Pi"', from=2-1, to=2-2] +\end{tikzcd}\] + +which follows from the definitions of $\Pi$ and $\la$. +\end{proof} + +\medskip + +\begin{lemma}\label{R_classifies} + Use $R$ to denote the fiber product + % https://q.uiver.app/#q=WzAsNCxbMSwwLCJcXFBvbHl7XFx0cH0ge1xcVHlwZX0iXSxbMSwxLCJcXFR5cGUiXSxbMCwwLCJSIl0sWzAsMSwiXFxUZXJtIl0sWzAsMSwiXFx0cF8qIFxcVGVybV4qIFxcVHlwZSJdLFsyLDAsIlxccmhvX1xcUG9seXt9Il0sWzIsMywiXFx0cF4qIFxcdHBfKiBcXFRlcm1eKiBcXFR5cGU9IFxccmhvX1xcVGVybSIsMl0sWzMsMSwiXFx0cCIsMl0sWzIsMSwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d +\[\begin{tikzcd} + R & {\Poly{\tp}{\Type}} \\ + \Term & \Type + \arrow["{\rho_{\Poly{}}}", from=1-1, to=1-2] + \arrow["{\tp^* \tp_* \Term^* \Type = \rho_\Term}"', from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow["{\tp_* \Term^* \Type}", from=1-2, to=2-2] + \arrow["\tp"', from=2-1, to=2-2] +\end{tikzcd}\] + + By the universal property of pullbacks, + The data of a map from a respresentable $\ep : \Ga \to R$ + corresponds to the data of + $\al : \Ga \to \Term$ and $(U \circ \al,B) : \Ga \to \Poly{\tp}{\Type}$. + Then by \cref{tp_classifies} this corresponds to the data of + $\al : \Ga \to \Term$ and $B : \Ga \cdot U \circ \al \to \Type$. + % https://q.uiver.app/#q=WzAsNSxbMiwxLCJcXFBvbHl7XFx0cH0ge1xcVHlwZX0iXSxbMiwyLCJcXFR5cGUiXSxbMSwxLCJSIl0sWzEsMiwiXFxUZXJtIl0sWzAsMCwiXFxHYSJdLFswLDEsIlxcdHBfKiBcXFRlcm1eKiBcXFR5cGUiXSxbMiwwLCJcXHJob19cXFBvbHl7fSJdLFsyLDMsIlxcdHBeKiBcXHRwXyogXFxUZXJtXiogXFxUeXBlPSBcXHJob19cXFRlcm0iLDJdLFszLDEsIlxcdHAiLDJdLFsyLDEsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFs0LDAsIihVIFxcY2lyYyBcXGFsLEIpIl0sWzQsMywiXFxhbCIsMl0sWzQsMiwiKFxcYWwsQikiLDFdXQ== +\[\begin{tikzcd} + \Ga \\ + & R & {\Poly{\tp} {\Type}} \\ + & \Term & \Type + \arrow["{(\al,B)}"{description}, from=1-1, to=2-2] + \arrow["{(U \circ \al,B)}"{description}, bend left, from=1-1, to=2-3] + \arrow["\al"', bend right, from=1-1, to=3-2] + \arrow["{\rho_{\Poly{}}}", from=2-2, to=2-3] + \arrow["{\rho_\Term}"', from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["{\tp_* \Term^* \Type}", from=2-3, to=3-3] + \arrow["\tp"', from=3-2, to=3-3] +\end{tikzcd}\] + + Precomposition by a substitution $\si : \De \to \Ga$ then act on such a pair by + \[ (\al, B) \mapsto (\al \circ \si, B \circ \si_{U \circ \al})\] +\end{lemma} + +\medskip + +\begin{defn}[Evaluation] + Define the operation of evaluation $\ev{\al}{B}$ to take + $\al : \Ga \to \ptgrpd$ and $B : \Ga \cdot U \circ \al \to \grpd$ + and return $\ev{\al}{B} : \Ga \to \grpd$, described below. +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + \Ga \\ + & R & {\Poly{\tp}{\Type}} \\ + {\Type \times \Term} & \Term & \Type \\ + \Type & \terminal + \arrow["{(\al,B)}"{description, pos=0.7}, from=1-1, to=2-2] + \arrow["{(A,B)}"{description}, bend left, from=1-1, to=2-3] + \arrow["{(\ev{\al}{B}, \al)}"{description}, dashed, from=1-1, to=3-1] + \arrow["\al"{description, pos=0.2}, from=1-1, to=3-2] + \arrow["{\ev{\al}{B}}"{description}, bend right = 50, dashed, from=1-1, to=4-1] + \arrow[from=2-2, to=2-3] + \arrow["\counit"{description, pos=0.7}, from=2-2, to=3-1] + \arrow[from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["{\tp_* \Term^* \Type}", from=2-3, to=3-3] + \arrow[from=3-1, to=3-2] + \arrow[from=3-1, to=4-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-1, to=4-2] + \arrow["\tp"', from=3-2, to=3-3] + \arrow[from=3-2, to=4-2] + \arrow[from=4-1, to=4-2] +\end{tikzcd}\] + where we write $A := U \circ \al$ and treat a map + $\Ga \to \grpd$ + as the same + as a map + $\Ga \to \Type$. + + More concisely, evaluation is a natural transformation $\ev{}{} : R \to \Type$, given by + \[ \ev{}{} = \pi_{\Type} \circ \counit \] +\end{defn} + +\medskip + +\begin{lemma}\label{evaluation_computation} + The functor $\ev{\al}{B} : \Ga \to \grpd$ can be computed as + \[ \ev{\al}{B} = B \circ a \] + where + % https://q.uiver.app/#q=WzAsNSxbMiwyLCJcXGdycGQiXSxbMiwxLCJcXHB0Z3JwZCJdLFsxLDIsIlxcR2EiXSxbMSwxLCJcXEdhIFxcY2RvdCBBIl0sWzAsMCwiXFxHYSJdLFsxLDBdLFsyLDAsIkEiLDJdLFszLDIsIlxcZGlzcHtBfSIsMl0sWzMsMV0sWzQsMiwiIiwwLHsibGV2ZWwiOjIsInN0eWxlIjp7ImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbNCwxLCJcXGFsIl0sWzQsMywiYSIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFszLDAsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ== +\[\begin{tikzcd} + \Ga \\ + & {\Ga \cdot A} & \ptgrpd \\ + & \Ga & \grpd + \arrow["a"{description}, dashed, from=1-1, to=2-2] + \arrow["\al", bend left, from=1-1, to=2-3] + \arrow[Rightarrow, no head, bend right, from=1-1, to=3-2] + \arrow[from=2-2, to=2-3] + \arrow["{\disp{A}}"', from=2-2, to=3-2] + \arrow[from=2-3, to=3-3] + \arrow["A"', from=3-2, to=3-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] +\end{tikzcd}\] +\end{lemma} +\begin{proof} + Since $\counit = (\ev{}{}, \rho_{\Term}) : R \to \Type$, + it suffices to find out how the counit computes. + The adjunction $\tp^{*} \dashv \tp_{*}$ + suggests that we use the way + \[ \widetilde{\counit} = \id_{\Poly{\tp}\Type}\] + computes. Namely for any $A : \Ga \to \grpd$ and $B : \Ga \cdot A \to \grpd$ + \begin{equation} + \widetilde{\counit} \circ (A, B) = (A, B) : \Ga \to \Poly{\tp}\Type + \label{evaluation_computation_equation1} + \end{equation} + + Working on both sides of \cref{evaluation_computation_equation1} we get + \begin{align*} + & (\ev{\var_{A}}{B} \circ U^{*}\disp{A}, \var_{A}) \\ + = \, & (\ev{}{}, \rho_{\Type}) \circ (\var_{A}, B \circ U^{*}(\disp{A})) \\ + = \, & (\ev{}{}, \rho_{\Type}) \circ \tp^{*} (A, B) + & \cref{evaluation_computation_diagram1} \\ + = \, & \counit \circ \tp^{*} (A, B) \\ + = \, & \overline{\widetilde{\counit} \circ (A, B)} \\ + = \, & \overline{(A, B)} \\ + = \, & (B, \var_{A}) + \end{align*} + Hence we know that evaluation of $B$ + (weakened to the context $\Ga \cdot A \cdot A$) + on a variable of type $A$ + is just $B$. + \[ \ev{\var_{A}}{B} \circ U^{*}\disp{A} = B\] + % diagrams TODO + + Then the naturality square for the natural transformation + $\ev{}{} : R \to \Type$ on $a : \Ga \to \Ga \cdot A$ + tells us that + \begin{align*} + & \ev{\al}{B} \\ + = \, & \ev{\Ga}{} (\al, B) \\ + = \, & \ev{\Ga}{} (\var_{A} \circ a, B \circ \U^{*} (\id_{\Ga}) \\ + = \, & \ev{\Ga}{} (\var_{A} \circ a, B \circ \U^{*} (\disp{A} \circ a)) \\ + = \, & \ev{\Ga}{} (\var_{A} \circ a, B \circ \U^{*} \disp{A} \circ U^{*} a) \\ + = \, & \ev{\Ga}{} ((\var_{A}, B \circ \U^{*} \disp{A}) \circ a) \\ + = \, & (\ev{\Ga \cdot A}{} (\var_{A}, B \circ \U^{*} \disp{A}) \circ a + & \text{by naturality}\\ + = \, & (\ev{\var_{A}}{B \circ \U^{*} \disp{A}}) \circ a \\ + = \, & B \circ a + \end{align*} + + + \begin{figure} +\centering +\begin{subfigure}{.5\textwidth} + \centering +% https://q.uiver.app/#q=WzAsNixbMSwxLCJcXFBvbHl7XFx0cH0ge1xcVHlwZX0iXSxbMSwyLCJcXFR5cGUiXSxbMCwxLCJSIl0sWzAsMiwiXFxUZXJtIl0sWzEsMCwiXFxHYSJdLFswLDAsIlxcR2EgXFxjZG90IEEiXSxbMCwxLCJcXHRwXyogXFxUZXJtXiogXFxUeXBlIiwxXSxbMiwwLCJcXHJob197XFxQb2x5e319Il0sWzIsMywiXFxyaG9fXFxUZXJtIiwxXSxbMywxLCJcXHRwIiwyXSxbMiwxLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XSxbNCwwLCIoQSxCKSIsMV0sWzUsNCwiXFxkaXNwe0F9Il0sWzUsMiwiXFx0cF4qKEEsQikiLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNSwzLCJcXHZhcl9BIiwyLHsiY3VydmUiOjR9XSxbNCwxLCJBIiwwLHsiY3VydmUiOi00fV1d +\begin{tikzcd} + {\Ga \cdot A} & \Ga \\ + R & {\Poly{\tp} {\Type}} \\ + \Term & \Type + \arrow["{\disp{A}}", from=1-1, to=1-2] + \arrow["{\tp^*(A,B)}"{description}, dashed, from=1-1, to=2-1] + \arrow["{\var_A}"', bend right = 60, from=1-1, to=3-1] + \arrow["{(A,B)}"{description}, from=1-2, to=2-2] + \arrow["A", bend left = 60, from=1-2, to=3-2] + \arrow["{\rho_{\Poly{}}}", from=2-1, to=2-2] + \arrow["{\rho_\Term}"{description}, from=2-1, to=3-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-1, to=3-2] + \arrow["{\tp_* \Term^* \Type}"{description}, from=2-2, to=3-2] + \arrow["\tp"', from=3-1, to=3-2] +\end{tikzcd} + \caption{$\tp^{*}(A,B)$} +\end{subfigure}% +\begin{subfigure}{.5\textwidth} + \centering + \begin{align*} + & \rho_{\Poly{}} \circ \tp^*(A,B) \\ + = \, & (A, B) \circ \disp{A} \\ + = \, & (A \circ \disp{A}, B \circ U^{*}\disp{A}) \\ + \text{Hence} + \end{align*} + \caption{$\tp^{*}(A,B) = (\var_{A}, B \circ U^{*}\disp{A})$} + \label{evaluation_computation_diagram1} +\end{subfigure} +\end{figure} + % (\var_{A}, B \circ \tp^* \disp_A) + + +\end{proof} + +\medskip + +\begin{defn}[Classifier for dependent pairs]\label{dependent_pair_classifier} + Recall the following definition of composition of polynomial endofunctors, + specialized to our situation +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + Q & R & R & {\Poly{\tp} {\Type}} \\ + {\Term \times \Term} & {\Type \times \Term} & \Term & \Type \\ + \Term & \Type & \terminal + \arrow[from=1-1, to=1-2] + \arrow["{\tp \triangleleft \tp}"{description}, bend left, from=1-1, to=1-4] + \arrow[from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow[Rightarrow, no head, from=1-2, to=1-3] + \arrow["\counit"{description}, from=1-2, to=2-2] + \arrow[from=1-3, to=1-4] + \arrow[from=1-3, to=2-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-3, to=2-4] + \arrow["{\tp_* \Term^* \Type}", from=1-4, to=2-4] + \arrow[from=2-1, to=2-2] + \arrow[from=2-1, to=3-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-1, to=3-2] + \arrow[from=2-2, to=2-3] + \arrow[from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["\tp"', from=2-3, to=2-4] + \arrow[from=2-3, to=3-3] + \arrow["\tp"', from=3-1, to=3-2] + \arrow[from=3-2, to=3-3] +\end{tikzcd}\] + + By the universal property of pullbacks, + a data of a map with representable domain $\ep : \Ga \to Q$ corresponds to the + data of a triple of maps $\al, \be : \Ga \to \Term$ + and $(A,B) : \Ga \to \Poly{\tp} \Type$ such that + $\tp \circ \be = \pi_{\Type} \circ \counit \circ (\al, B)$ + and $A = \tp \circ \al$. +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + \Ga \\ + & Q & R & R & {\Poly{\tp} {\Type}} \\ + & {\Term \times \Term} & {\Type \times \Term} & \Term & \Type \\ + & \Term & \Type & \terminal + \arrow[from=2-2, to=2-3] + \arrow[from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow[""{name=0, anchor=center, inner sep=0}, Rightarrow, no head, from=2-3, to=2-4] + \arrow["\counit"{description}, from=2-3, to=3-3] + \arrow[from=2-4, to=2-5] + \arrow[from=2-4, to=3-4] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-4, to=3-5] + \arrow["{\tp_* \Term^* \Type}", from=2-5, to=3-5] + \arrow[from=3-2, to=3-3] + \arrow[from=3-2, to=4-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-2, to=4-3] + \arrow[from=3-3, to=3-4] + \arrow[from=3-3, to=4-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-3, to=4-4] + \arrow["\tp"', from=3-4, to=3-5] + \arrow[from=3-4, to=4-4] + \arrow["\tp"', from=4-2, to=4-3] + \arrow[from=4-3, to=4-4] + \arrow["\ep"{description}, from=1-1, to=2-2] + \arrow["{(A,B)}"{description}, bend left, from=1-1, to=2-5] + \arrow["\al"{description, pos=0.2}, bend left = 10, from=1-1, to=3-4] + \arrow["\be"', from=1-1, bend right, to=4-2] + \arrow["{( \al , B ) }"{description}, bend left = 25, from=1-1, to=2-3] +\end{tikzcd}\] + + + This in turn corresponds to three functors $\al, \be : \Ga \to \ptgrpd$ and + $B : \Ga \cdot U \circ \al \to \grpd$, + such that $U \circ \be = \ev{\al} B$. + So we will write + \[\ep = (\be, \al, B)\] + {\color{gray} + Type theoretically $\al = (A , a : A)$ and $\ev{\al} B = B a$ and + $\be = (B a, b : B a)$. + Then composing $\ep$ with $\tp \triangleleft \tp$ returns $\ga$, + which consists of $(A, B)$. + It is in this sense that $Q$ classifies pairs of dependent terms, + and $\tp \triangleleft \tp$ extracts the underlying types.} +\end{defn} + +\medskip + +\begin{defn}[Interpretation of $\Si$] + We define the natural transformation + \[\Si : \Poly{\tp} \Type \to \Type\] + as that which is induced (\cref{BC_iff_nat_trans}) by the $\Si$-former operation (\cref{pi_classifier_op}). + + To define $\pair : Q \to \Term$, let $\Ga$ be a groupoid + % \begin{align*} + % & Q \, \Ga \\ + % \iso & + % \{ (\be : \ga \to \ptgrpd, \al : \Ga \to \ptgrpd, + % B : \Ga \cdot U \circ \al \to \grpd) \st U \circ \be = \ev{\al}{\be} \} + % \\ + % \to & [\Ga, \ptgrpd] + % \end{align*} + and $(\be,\al,B) : \Ga \to Q$ (such that $U \circ \be = \ev{\al}{\be}$). + We define a functor $\pair_{\Ga}(\be,\al,B) : \Ga \to \ptgrpd$ such that + on objects $x \in \Ga$, the functor returns $(\Si_{A} B \, x, (a_{x}, b_{a_{x}}))$, + where + (using \cref{evaluation_computation} + $U \circ \be x = \ev{\al}{B} x = B(x, a_{x})$) + \[\al \, x = (A \, x, a_{x}) \quad \text{ and } \quad \be \, x = (B (x , a_{x}), b_{a_{x}})\] + and on morphisms $f : x \to y$, the functor returns $(\Si_{A} B \, f, (\phi_{f}, \psi_{f}))$, + where (using \cref{evaluation_computation} + $U \circ \be f = \ev{\al}{B} f = B(f, \phi_{f})$) + \[\al \, f = (A \, f, \phi_{f} { \color{gray} : A \, f \, a_{x} \to a_{y}}) + \quad \text{ and } \quad + \be \, f = (B(f, \phi_{f}), \psi_{f} {\color{gray}: B(f, \phi_{f}) \, b_{a_{x}}\to b_{a_{y}}}) + \] + + $\Si$ and $\pair$ combine to give us a pullback square +% https://q.uiver.app/#q=WzAsNCxbMCwwLCJRIl0sWzAsMSwiXFxQb2x5e1xcdHB9e1xcVHlwZX0iXSxbMSwwLCJcXFRlcm0iXSxbMSwxLCJcXFR5cGUiXSxbMCwxLCJcXHRwIFxcdHJpYW5nbGVsZWZ0IFxcdHAiLDJdLFswLDIsIlxccGFpciJdLFsyLDMsIlxcdHAiXSxbMSwzLCJcXFNpIiwyXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0= +\[\begin{tikzcd} + Q & \Term \\ + {\Poly{\tp}{\Type}} & \Type + \arrow["\pair", from=1-1, to=1-2] + \arrow["{\tp \triangleleft \tp}"', from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow["\tp", from=1-2, to=2-2] + \arrow["\Si"', from=2-1, to=2-2] +\end{tikzcd}\] +\end{defn} +\begin{proof} + To show naturality of $\pair$, + suppose $\si : \De \to \Ga$ is a functor between groupoids. +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + {\Pshgrpd(\De,Q)} &&& {[\De,\ptgrpd]} \\ + & {(\be \circ \si,\al \circ \si,B \circ \si_A)} & {?} \\ + & {(\be,\al,B)} & {\pair_\Ga(\be,\al,B)} \\ + {\Pshgrpd(\Ga,Q)} &&& {[\Ga,\ptgrpd]} + \arrow["{\pair_\De}", from=1-1, to=1-4] + \arrow[maps to, from=2-2, to=2-3] + \arrow[maps to, from=3-2, to=2-2] + \arrow[maps to, from=3-2, to=3-3] + \arrow[maps to, from=3-3, to=2-3] + \arrow["{\circ \si}", from=4-1, to=1-1] + \arrow["{\pair_\Ga}"', from=4-1, to=4-4] + \arrow["{\circ \si}"', from=4-4, to=1-4] +\end{tikzcd}\] + + So we check that for any $x \in \Ga$, + \begin{align*} + & \pair_\De (\be \circ \si, \al \circ \si, B \circ \si_A) \, x \\ + = \, & (\Si_{A \circ \si} B \circ \si_A \, x, (a_x, b_{a_x})) \\ + = \, & ((\Si_{A} B) \circ \si \, x, (a_x, b_{a_x})) \\ + = \, & \pair_\Ga (\be, \al, B) \circ \si \, x + \end{align*} + where + \[ \al \circ \si \, x = (A \circ \si x, a_{x}) + \quad + \text{and} + \quad + \be \circ \si \, x = (\ev{\al}{B} \circ \si \, x, b_{a_{x}}) + \] + and so on. + + It follows from the definition of $\pair$ that the square commutes. + To show that it is pullback, it suffices to show that for each $\Ga$, + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXFBzaGdycGQgKFxcR2EsIFEpIl0sWzAsMSwiXFxQc2hncnBkKFxcR2EsXFxQb2x5e1xcdHB9e1xcVHlwZX0pIl0sWzEsMCwiW1xcR2EsXFxwdGdycGRdIl0sWzEsMSwiW1xcR2EsXFxncnBkXSJdLFswLDEsIlxcdHAgXFx0cmlhbmdsZWxlZnQgXFx0cCBcXGNpcmMgLSIsMl0sWzAsMiwiXFxwYWlyX1xcR2EiXSxbMiwzLCJVIFxcY2lyYyAtIl0sWzEsMywiXFxTaV9cXEdhIiwyXV0= +\[\begin{tikzcd} + {\Pshgrpd (\Ga, Q)} & {[\Ga,\ptgrpd]} \\ + {\Pshgrpd(\Ga,\Poly{\tp}{\Type})} & {[\Ga,\grpd]} + \arrow["{\pair_\Ga}", from=1-1, to=1-2] + \arrow["{\tp \triangleleft \tp \circ -}"', from=1-1, to=2-1] + \arrow["{U \circ -}", from=1-2, to=2-2] + \arrow["{\Si_\Ga}"', from=2-1, to=2-2] +\end{tikzcd}\] + is a pullback. + Since we are in $\Set$, + it suffices to just show the universal property applied to a point: + so for any $A : \Ga \to \grpd$, any $B : \Ga \cdot A \to \grpd$, + and any $p : \Ga \to \ptgrpd$, + such that + \[ U \circ p = \Si_{\Ga} (A, B)\] + there exists a unique $(\be, \al, B) : \Ga \to Q$ + such that + \[ \pair_{\Ga}(\be,\al,B) = p + \quad + \text{and} + \quad + \tp \triangleleft \tp \circ (B,\al, B) = (A, B) + \] + Indeed if we write + \[ p \, x = (\Si_{A} B \, x , + (a_{x} {\color{gray} \in A x}, b_{x} {\color{gray} \in B(x, a_{x})}))\] + this uniquely determines $\al$ and $\be$ as + \[ \al \, x = (A x, a_{x}) + \quad + \text{and} + \quad + \be \, x = (\ev{\al}{B} \, x, b_{x}) + \] + + + +\end{proof} diff --git a/blueprint/src/macros/common.tex b/blueprint/src/macros/common.tex index 83d71e1..15060b4 100644 --- a/blueprint/src/macros/common.tex +++ b/blueprint/src/macros/common.tex @@ -9,10 +9,99 @@ % (this is what the [theorem] arguments mean) and never resets it. % If you want for instance to number them within chapters then you can add % [chapter] at the end of the next line. -\newtheorem{theorem}{Theorem} -\newtheorem{proposition}[theorem]{Proposition} + +%%%%%%% THEOREMS %%%%%%%%% +\newtheorem{theorem}{Theorem}[section] +\newtheorem*{theorem*}{Theorem} +\newtheorem{prop}[theorem]{Proposition} +\newtheorem{obs}[theorem]{Observation} \newtheorem{lemma}[theorem]{Lemma} -\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{cor}[theorem]{Corollary} +\newtheorem{applemma}{Lemma}[section] \theoremstyle{definition} -\newtheorem{definition}[theorem]{Definition} \ No newline at end of file +\newtheorem{defn}[theorem]{Definition} + +\theoremstyle{remark} +\newtheorem{rmk}[theorem]{Remark} +\newtheorem*{eg}{Example} +\newtheorem{ex}{Exercise} +\newtheorem*{remark*}{Remark} +\newtheorem*{remarks*}{Remarks} +\newtheorem*{notation*}{Notation} +\newtheorem*{convention*}{Convention} + +\newenvironment{comment}{\begin{quote} \em Comment. }{\end{quote}} + +%%%%%%% SYMBOLS %%%%%%%%% + +\newcommand{\defeq}{=_{\mathrm{def}}} +\newcommand{\iso}{\cong} +\newcommand{\equi}{\simeq} +\newcommand{\retequi}{\unlhd} +\newcommand{\op}{\mathrm{op}} +\newcommand{\Id}{\mathrm{Id}} +\newcommand{\Nat}{\mathbb{N}} +\newcommand{\co}{\colon} +\newcommand{\st}{\,|\,} + +% \newcommand{\cat}{\mathbb} +\newcommand{\catC}{\mathbb{C}} +\newcommand{\tcat}{\mathbf} +\newcommand{\id}{\mathsf{id}} +\newcommand{\psh}[1]{\mathbf{Psh}(#1)} %% consider renaming to \Psh +\newcommand{\yo}{\mathsf{y}} +\newcommand{\pshC}{\psh{\catC}} +\newcommand{\Type}{\mathsf{Ty}} +\newcommand{\Term}{\mathsf{Tm}} +\newcommand{\tp}{\mathsf{tp}} +\newcommand{\disp}[1]{\mathsf{disp}_{#1}} +\newcommand{\var}{\mathsf{var}} +\newcommand{\Prop}{\mathsf{Prop}} +\newcommand{\U}{\mathsf{U}} +\newcommand{\E}{\mathsf{E}} +\newcommand{\El}{\mathsf{El}} +\newcommand{\Poly}[1]{\mathsf{Poly}_{#1}} +\newcommand{\pair}{\mathsf{pair}} +\newcommand{\fst}{\mathsf{fst}} +\newcommand{\ev}[2]{\mathsf{ev}_{#1} \, #2} +\newcommand{\counit}{\mathsf{counit}} + +\newcommand{\Fib}{\mathsf{Fib}} +\newcommand{\lift}[2]{\mathsf{lift} \, #1 \, #2} +\newcommand{\fiber}{\mathsf{fiber}} +\newcommand{\terminal}{\bullet} +\newcommand{\Two}{\bullet+\bullet} +\newcommand{\Interval}{\mathbb{I}} + +\newcommand{\set}{\tcat{set}} +\newcommand{\Set}{\tcat{Set}} +\newcommand{\cat}{\tcat{cat}} +\newcommand{\ptcat}{\tcat{cat}_\bullet} +\newcommand{\Cat}{\tcat{Cat}} +\newcommand{\ptCat}{\tcat{Cat}_\bullet} +\newcommand{\grpd}{\tcat{grpd}} +% \newcommand{\Grpd}{\tcat{Grpd}} +\newcommand{\ptgrpd}{\tcat{grpd}_\bullet} +% \newcommand{\ptGrpd}{\tcat{Grpd}_\bullet} +\newcommand{\Pshgrpd}{\mathbf{Psh}(\grpd)} +\newcommand{\PshCat}{\mathbf{Psh}(\Cat)} + +% % Greek +\newcommand{\al}{\alpha} +\newcommand{\be}{\beta} +\newcommand{\ga}{\gamma} +\newcommand{\de}{\delta} +\newcommand{\ep}{\varepsilon} +\newcommand{\io}{\iota} +\newcommand{\ka}{\kappa} +\newcommand{\la}{\lambda} +\newcommand{\om}{\omega} +\newcommand{\si}{\sigma} + +\newcommand{\Ga}{\Gamma} +\newcommand{\De}{\Delta} +\newcommand{\Th}{\Theta} +\newcommand{\La}{\Lambda} +\newcommand{\Si}{\Sigma} +\newcommand{\Om}{\Omega} diff --git a/blueprint/src/macros/prooftree.tex b/blueprint/src/macros/prooftree.tex new file mode 100644 index 0000000..abb7d44 --- /dev/null +++ b/blueprint/src/macros/prooftree.tex @@ -0,0 +1,347 @@ +\message{} +%% Build proof tree for Natural Deduction, Sequent Calculus, etc. +%% WITH SHORTENING OF PROOF RULES! +%% Paul Taylor, begun 10 Oct 1989 +%% *** THIS IS ONLY A PRELIMINARY VERSION AND THINGS MAY CHANGE! *** +%% +%% 2 Aug 1996: fixed \mscount and \proofdotnumber +%% +%% \prooftree +%% hyp1 produces: +%% hyp2 +%% hyp3 hyp1 hyp2 hyp3 +%% \justifies -------------------- rulename +%% concl concl +%% \thickness=0.08em +%% \shiftright 2em +%% \using +%% rulename +%% \endprooftree +%% +%% where the hypotheses may be similar structures or just formulae. +%% +%% To get a vertical string of dots instead of the proof rule, do +%% +%% \prooftree which produces: +%% [hyp] +%% \using [hyp] +%% name . +%% \proofdotseparation=1.2ex .name +%% \proofdotnumber=4 . +%% \leadsto . +%% concl concl +%% \endprooftree +%% +%% Within a prooftree, \[ and \] may be used instead of \prooftree and +%% \endprooftree; this is not permitted at the outer level because it +%% conflicts with LaTeX. Also, +%% \Justifies +%% produces a double line. In LaTeX you can use \begin{prooftree} and +%% \end{prootree} at the outer level (however this will not work for the inner +%% levels, but in any case why would you want to be so verbose?). +%% +%% All of of the keywords except \prooftree and \endprooftree are optional +%% and may appear in any order. They may also be combined in \newcommand's +%% eg "\def\Cut{\using\sf cut\thickness.08em\justifies}" with the abbreviation +%% "\prooftree hyp1 hyp2 \Cut \concl \endprooftree". This is recommended and +%% some standard abbreviations will be found at the end of this file. +%% +%% \thickness specifies the breadth of the rule in any units, although +%% font-relative units such as "ex" or "em" are preferable. +%% It may optionally be followed by "=". +%% \proofrulebreadth=.08em or \setlength\proofrulebreadth{.08em} may also be +%% used either in place of \thickness or globally; the default is 0.04em. +%% \proofdotseparation and \proofdotnumber control the size of the +%% string of dots +%% +%% If proof trees and formulae are mixed, some explicit spacing is needed, +%% but don't put anything to the left of the left-most (or the right of +%% the right-most) hypothesis, or put it in braces, because this will cause +%% the indentation to be lost. +%% +%% By default the conclusion is centered wrt the left-most and right-most +%% immediate hypotheses (not their proofs); \shiftright or \shiftleft moves +%% it relative to this position. (Not sure about this specification or how +%% it should affect spreading of proof tree.) +% +% global assignments to dimensions seem to have the effect of stretching +% diagrams horizontally. +% +%%========================================================================== + +\def\introrule{{\cal I}}\def\elimrule{{\cal E}}%% +\def\andintro{\using{\land}\introrule\justifies}%% +\def\impelim{\using{\Rightarrow}\elimrule\justifies}%% +\def\allintro{\using{\forall}\introrule\justifies}%% +\def\allelim{\using{\forall}\elimrule\justifies}%% +\def\falseelim{\using{\bot}\elimrule\justifies}%% +\def\existsintro{\using{\exists}\introrule\justifies}%% + +%% #1 is meant to be 1 or 2 for the first or second formula +\def\andelim#1{\using{\land}#1\elimrule\justifies}%% +\def\orintro#1{\using{\lor}#1\introrule\justifies}%% + +%% #1 is meant to be a label corresponding to the discharged hypothesis/es +\def\impintro#1{\using{\Rightarrow}\introrule_{#1}\justifies}%% +\def\orelim#1{\using{\lor}\elimrule_{#1}\justifies}%% +\def\existselim#1{\using{\exists}\elimrule_{#1}\justifies} + +%%========================================================================== + +\newdimen\proofrulebreadth \proofrulebreadth=.05em +\newdimen\proofdotseparation \proofdotseparation=1.25ex +\newdimen\proofrulebaseline \proofrulebaseline=2ex +\newcount\proofdotnumber \proofdotnumber=3 +\let\then\relax +\def\hfi{\hskip0pt plus.0001fil} +\mathchardef\squigto="3A3B +% +% flag where we are +\newif\ifinsideprooftree\insideprooftreefalse +\newif\ifonleftofproofrule\onleftofproofrulefalse +\newif\ifproofdots\proofdotsfalse +\newif\ifdoubleproof\doubleprooffalse +\let\wereinproofbit\relax +% +% dimensions and boxes of bits +\newdimen\shortenproofleft +\newdimen\shortenproofright +\newdimen\proofbelowshift +\newbox\proofabove +\newbox\proofbelow +\newbox\proofrulename +% +% miscellaneous commands for setting values +\def\shiftproofbelow{\let\next\relax\afterassignment\setshiftproofbelow\dimen0 } +\def\shiftproofbelowneg{\def\next{\multiply\dimen0 by-1 }% +\afterassignment\setshiftproofbelow\dimen0 } +\def\setshiftproofbelow{\next\proofbelowshift=\dimen0 } +\def\setproofrulebreadth{\proofrulebreadth} + +%============================================================================= +\def\prooftree{% NESTED ZERO (\ifonleftofproofrule) +% +% first find out whether we're at the left-hand end of a proof rule +\ifnum \lastpenalty=1 +\then \unpenalty +\else \onleftofproofrulefalse +\fi +% +% some space on left (except if we're on left, and no infinity for outermost) +\ifonleftofproofrule +\else \ifinsideprooftree + \then \hskip.5em plus1fil + \fi +\fi +% +% begin our proof tree environment +\bgroup% NESTED ONE (\proofbelow, \proofrulename, \proofabove, +% \shortenproofleft, \shortenproofright, \proofrulebreadth) +\setbox\proofbelow=\hbox{}\setbox\proofrulename=\hbox{}% +\let\justifies\proofover\let\leadsto\proofoverdots\let\Justifies\proofoverdbl +\let\using\proofusing\let\[\prooftree +\ifinsideprooftree\let\]\endprooftree\fi +\proofdotsfalse\doubleprooffalse +\let\thickness\setproofrulebreadth +\let\shiftright\shiftproofbelow \let\shift\shiftproofbelow +\let\shiftleft\shiftproofbelowneg +\let\ifwasinsideprooftree\ifinsideprooftree +\insideprooftreetrue +% +% now begin to set the top of the rule (definitions local to it) +\setbox\proofabove=\hbox\bgroup$\displaystyle % NESTED TWO +\let\wereinproofbit\prooftree +% +% these local variables will be copied out: +\shortenproofleft=0pt \shortenproofright=0pt \proofbelowshift=0pt +% +% flags to enable inner proof tree to detect if on left: +\onleftofproofruletrue\penalty1 +} + +%============================================================================= +% end whatever box and copy crucial values out of it +\def\eproofbit{% NESTED TWO +% +% various hacks applicable to hypothesis list +\ifx \wereinproofbit\prooftree +\then \ifcase \lastpenalty + \then \shortenproofright=0pt % 0: some other object, no indentation + \or \unpenalty\hfil % 1: empty hypotheses, just glue + \or \unpenalty\unskip % 2: just had a tree, remove glue + \else \shortenproofright=0pt % eh? + \fi +\fi +% +% pass out crucial values from scope +\global\dimen0=\shortenproofleft +\global\dimen1=\shortenproofright +\global\dimen2=\proofrulebreadth +\global\dimen3=\proofbelowshift +\global\dimen4=\proofdotseparation +\global\count255=\proofdotnumber +% +% end the box +$\egroup % NESTED ONE +% +% restore the values +\shortenproofleft=\dimen0 +\shortenproofright=\dimen1 +\proofrulebreadth=\dimen2 +\proofbelowshift=\dimen3 +\proofdotseparation=\dimen4 +\proofdotnumber=\count255 +} + +%============================================================================= +\def\proofover{% NESTED TWO +\eproofbit % NESTED ONE +\setbox\proofbelow=\hbox\bgroup % NESTED TWO +\let\wereinproofbit\proofover +$\displaystyle +}% +% +%============================================================================= +\def\proofoverdbl{% NESTED TWO +\eproofbit % NESTED ONE +\doubleprooftrue +\setbox\proofbelow=\hbox\bgroup % NESTED TWO +\let\wereinproofbit\proofoverdbl +$\displaystyle +}% +% +%============================================================================= +\def\proofoverdots{% NESTED TWO +\eproofbit % NESTED ONE +\proofdotstrue +\setbox\proofbelow=\hbox\bgroup % NESTED TWO +\let\wereinproofbit\proofoverdots +$\displaystyle +}% +% +%============================================================================= +\def\proofusing{% NESTED TWO +\eproofbit % NESTED ONE +\setbox\proofrulename=\hbox\bgroup % NESTED TWO +\let\wereinproofbit\proofusing +\kern0.3em$ +} + +%============================================================================= +\def\endprooftree{% NESTED TWO +\eproofbit % NESTED ONE +% \dimen0 = length of proof rule +% \dimen1 = indentation of conclusion wrt rule +% \dimen2 = new \shortenproofleft, ie indentation of conclusion +% \dimen3 = new \shortenproofright, ie +% space on right of conclusion to end of tree +% \dimen4 = space on right of conclusion below rule + \dimen5 =0pt% spread of hypotheses +% \dimen6, \dimen7 = height & depth of rule +% +% length of rule needed by proof above +\dimen0=\wd\proofabove \advance\dimen0-\shortenproofleft +\advance\dimen0-\shortenproofright +% +% amount of spare space below +\dimen1=.5\dimen0 \advance\dimen1-.5\wd\proofbelow +\dimen4=\dimen1 +\advance\dimen1\proofbelowshift \advance\dimen4-\proofbelowshift +% +% conclusion sticks out to left of immediate hypotheses +\ifdim \dimen1<0pt +\then \advance\shortenproofleft\dimen1 + \advance\dimen0-\dimen1 + \dimen1=0pt +% now it sticks out to left of tree! + \ifdim \shortenproofleft<0pt + \then \setbox\proofabove=\hbox{% + \kern-\shortenproofleft\unhbox\proofabove}% + \shortenproofleft=0pt + \fi +\fi +% +% and to the right +\ifdim \dimen4<0pt +\then \advance\shortenproofright\dimen4 + \advance\dimen0-\dimen4 + \dimen4=0pt +\fi +% +% make sure enough space for label +\ifdim \shortenproofright<\wd\proofrulename +\then \shortenproofright=\wd\proofrulename +\fi +% +% calculate new indentations +\dimen2=\shortenproofleft \advance\dimen2 by\dimen1 +\dimen3=\shortenproofright\advance\dimen3 by\dimen4 +% +% make the rule or dots, with name attached +\ifproofdots +\then + \dimen6=\shortenproofleft \advance\dimen6 .5\dimen0 + \setbox1=\vbox to\proofdotseparation{\vss\hbox{$\cdot$}\vss}% + \setbox0=\hbox{% + \advance\dimen6-.5\wd1 + \kern\dimen6 + $\vcenter to\proofdotnumber\proofdotseparation + {\leaders\box1\vfill}$% + \unhbox\proofrulename}% +\else \dimen6=\fontdimen22\the\textfont2 % height of maths axis + \dimen7=\dimen6 + \advance\dimen6by.5\proofrulebreadth + \advance\dimen7by-.5\proofrulebreadth + \setbox0=\hbox{% + \kern\shortenproofleft + \ifdoubleproof + \then \hbox to\dimen0{% + $\mathsurround0pt\mathord=\mkern-6mu% + \cleaders\hbox{$\mkern-2mu=\mkern-2mu$}\hfill + \mkern-6mu\mathord=$}% + \else \vrule height\dimen6 depth-\dimen7 width\dimen0 + \fi + \unhbox\proofrulename}% + \ht0=\dimen6 \dp0=-\dimen7 +\fi +% +% set up to centre outermost tree only +\let\doll\relax +\ifwasinsideprooftree +\then \let\VBOX\vbox +\else \ifmmode\else$\let\doll=$\fi + \let\VBOX\vcenter +\fi +% this \vbox or \vcenter is the actual output: +\VBOX {\baselineskip\proofrulebaseline \lineskip.2ex + \expandafter\lineskiplimit\ifproofdots0ex\else-0.6ex\fi + \hbox spread\dimen5 {\hfi\unhbox\proofabove\hfi}% + \hbox{\box0}% + \hbox {\kern\dimen2 \box\proofbelow}}\doll% +% +% pass new indentations out of scope +\global\dimen2=\dimen2 +\global\dimen3=\dimen3 +\egroup % NESTED ZERO +\ifonleftofproofrule +\then \shortenproofleft=\dimen2 +\fi +\shortenproofright=\dimen3 +% +% some space on right and flag we've just made a tree +\onleftofproofrulefalse +\ifinsideprooftree +\then \hskip.5em plus 1fil \penalty2 +\fi +} + +%========================================================================== +% IDEAS +% 1. Specification of \shiftright and how to spread trees. +% 2. Spacing command \m which causes 1em+1fil spacing, over-riding +% exisiting space on sides of trees and not affecting the +% detection of being on the left or right. +% 3. Hack using \@currenvir to detect LaTeX environment; have to +% use \aftergroup to pass \shortenproofleft/right out. +% 4. (Pie in the sky) detect how much trees can be "tucked in" +% 5. Discharged hypotheses (diagonal lines). diff --git a/blueprint/src/natural_model.tex b/blueprint/src/natural_model.tex new file mode 100644 index 0000000..fea0a66 --- /dev/null +++ b/blueprint/src/natural_model.tex @@ -0,0 +1,115 @@ +\subsection{Types} + +Assume an inaccessible cardinal $\lambda$. Write $\Set$ for the category of all sets. Say that a set $A$ is $\lambda$-small if $|A| < \lambda$. Write $\Set_\lambda$ for the full +subcategory of $\Set$ spanned by $\lambda$-small sets. + +Let $\mathbb{C}$ be a small category, i.e.~a category whose class of objects is a set and +whose hom-classes are sets. +% We do not assume that $\mathbb{C}$ is $\lambda$-small for +% the moment. + +We write $\pshC$ for the category of presheaves over $\catC$, +\[ +\pshC \defeq [\catC^\op, \Set] +\] + +% Because of the assumption of the existence of $\lambda$, $\pshC$ has additional structure. Let +% \[ +% \Term \to \Type +% \] +% be the Hofmann-Streicher universe in $\pshC$ associated to $\lambda$. In particular, +% \[ +% \Type(c) \defeq \{ A \co (\catC_{/c})^\op \to \Set_{\lambda} \} +% \] + +The Natural Model associated to a presentable map $\tp \co \Term \to \Type$ consists of +\begin{itemize} +\item contexts as objects $\Gamma, \Delta, \ldots \in \catC$, +\item a type in context $\yo (\Gamma)$ as a map $A \co \yo(\Gamma) \to \Type$, +\item a term of type $A$ in context $\Gamma$ as a map $a \co \yo(\Gamma) \to \Term$ such that +\[ +\xymatrix{ + & \Term \ar[d]^{\tp} \\ +\Gamma \ar[r]_-A \ar[ur]^{a} & \Type} +\] +commutes, +\item an operation called ``context extension'' which given a context $\Gamma$ and a type $A \co \yo(\Gamma) \to \Type$ produces a context $\Gamma\cdot A$ which fits into a pullback diagram below. +\[ +\xymatrix{ +\yo(\Gamma.A) \ar[r] \ar[d] & \Term \ar[d] \\ +\yo(\Gamma) \ar[r]_-{A} & \Type} +\] +\end{itemize} + +% In the internal type theory of $\pshC$, we write +% \[ +% (\Gamma) \ A \co \Type \qquad (\Gamma) \ a \co A +% \] +% to mean that $A$ is a type in context $\Gamma$ and that $a$ is an element of type $A$ in context +% $\Gamma$, respectively. + + + + +{\bf Remark.} +Sometimes, we first construct a presheaf $X$ over $\Gamma$ and observe that it can be classified by a map into $\Type$. We write +\[ +\xymatrix@C=1cm{ +X \ar[r] \ar[d]& \Term \ar[d] \\ +\yo(\Gamma) \ar[r]_-{\ulcorner X \urcorner} & \Type} +\] +to express this situation, i.e.~$X \cong \yo(\Gamma \cdot \ulcorner X \urcorner)$. + +\medskip + + + +\subsection{A type of small types} + +We now wish to formulate a condition that allows us to have a type of small types, written $\U$, not just {\em judgement} expressing that something is a type. With this notation, the judgements that we would like to derive is +\[ + \U \co \Type \qquad + \begin{prooftree} + a \co \U + \justifies + \El(a) \co \Type + \end{prooftree} +\] + +(A sufficient and natural condition for this seems to be that we now have another inaccessible cardinal $\kappa$, with $\kappa < \lambda$.) + +In the Natural Model, a universe $\U$ is postulated by a map +\[ +\pi \co \E \to \U +\] + +In the Natural Model: +\begin{itemize} +\item There is a pullback diagram of the form +\[ +\xymatrix{ +\U \ar[r] \ar[d] & \Term \ar[d] \\ +1 \ar[r]_-{\ulcorner \U \urcorner } & \Type } +\] +\item There is an inclusion of $\U$ into $\Type$ +\[ +\El \co \U \rightarrowtail \Type +\] +\item $\pi : \E \to \U$ is obtained as pullback of $\tp$; There is a pullback diagram +\[ +\xymatrix{ +E \ar@{>->}[r] \ar[d] & \Term \ar[d] \\ +\U \ar@{>->}[r]_-{\El} & \Type } +\] + \end{itemize} + +With the notation above, we get +\[ +\xymatrix{ +\yo (\Gamma.\El(a)) \ar[r] \ar[d] & \E \ar[r] \ar[d] & \Term \ar[d] \\ +\yo (\Gamma) \ar[r]_a \ar@/_2pc/[rr]_-{A} & \U \ar[r]_{\El} & \Type} +\] +Both squares above are pullback squares. + + +\subsection{The Universe in Embedded Type Theory (HoTT0) and the relationship to the Natural Model} diff --git a/blueprint/src/print.tex b/blueprint/src/print.tex index 864c8e8..b10d7e8 100644 --- a/blueprint/src/print.tex +++ b/blueprint/src/print.tex @@ -16,18 +16,41 @@ \usepackage{expl3} -\usepackage{amssymb, amsthm, mathtools} +\usepackage{amsthm,amssymb,mathtools} \usepackage[unicode,colorlinks=true,linkcolor=blue,urlcolor=magenta, citecolor=blue]{hyperref} \usepackage[warnings-off={mathtools-colon,mathtools-overbracket}]{unicode-math} +\usepackage{amsmath,stmaryrd,enumerate,a4,array} +\usepackage{geometry} +\usepackage{url} +\usepackage[cmtip,all]{xy} +\usepackage{subcaption} +\usepackage{cleveref} +\numberwithin{equation}{section} +\usepackage[parfill]{parskip} +\usepackage{tikz, tikz-cd, float} % Commutative Diagrams + +\input{macros/prooftree} \input{macros/common} \input{macros/print} \title{A Groupoidal Natural Model of HoTT in Lean 4} -\author{Sina Hazratpour, Joseph Huan} +\author{Sina Hazratpour, Joseph Hua} \begin{document} \maketitle -\input{content} + + +\section{Natural Models} +\input{natural_model} + +\section{The Groupoid Model} +\input{groupoid_model} + + +\bibliography{refs.bib}{} +\bibliographystyle{alpha} + + \end{document} diff --git a/blueprint/src/refs.bib b/blueprint/src/refs.bib new file mode 100644 index 0000000..84b0cfe --- /dev/null +++ b/blueprint/src/refs.bib @@ -0,0 +1,30 @@ +@misc{awodey2023hofmannstreicheruniverses, + title={On Hofmann-Streicher universes}, + author={Steve Awodey}, + year={2023}, + eprint={2205.10917}, + archivePrefix={arXiv}, + primaryClass={math.CT}, + url={https://arxiv.org/abs/2205.10917}, +} + +@incollection {hofmannstreicher1996, + AUTHOR = {Hofmann, Martin and Streicher, Thomas}, + TITLE = {The groupoid interpretation of type theory}, + BOOKTITLE = {Twenty-five years of constructive type theory + ({V}enice, 1995)}, + SERIES = {Oxford Logic Guides}, + VOLUME = {36}, + PAGES = {83--111}, + PUBLISHER = {Oxford Univ. Press}, + ADDRESS = {New York}, + YEAR = {1998}, + MRCLASS = {03B15 (68N15 68Q55)}, + MRNUMBER = {1686862}, +} + +@misc{joyalnlabmodelstructuresoncat, + author = {André Joyal}, + title = {Model structures on Cat}, + howpublished = {\url{https://ncatlab.org/joyalscatlab/published/Model+structures+on+Cat}}, +} diff --git a/blueprint/src/web.bbl b/blueprint/src/web.bbl new file mode 100644 index 0000000..e67c156 --- /dev/null +++ b/blueprint/src/web.bbl @@ -0,0 +1,20 @@ +\begin{thebibliography}{Awo23} + +\bibitem[Awo23]{awodey2023hofmannstreicheruniverses} +Steve Awodey. +\newblock On hofmann-streicher universes, 2023. + +\bibitem[HS98]{hofmannstreicher1996} +Martin Hofmann and Thomas Streicher. +\newblock The groupoid interpretation of type theory. +\newblock In {\em Twenty-five years of constructive type theory ({V}enice, + 1995)}, volume~36 of {\em Oxford Logic Guides}, pages 83--111. Oxford Univ. + Press, New York, 1998. + +\bibitem[Joy]{joyalnlabmodelstructuresoncat} +André Joyal. +\newblock Model structures on cat. +\newblock + \url{https://ncatlab.org/joyalscatlab/published/Model+structures+on+Cat}. + +\end{thebibliography} diff --git a/blueprint/src/web.tex b/blueprint/src/web.tex index 3000e3a..435a933 100644 --- a/blueprint/src/web.tex +++ b/blueprint/src/web.tex @@ -23,5 +23,5 @@ \begin{document} \maketitle -\input{content} -\end{document} \ No newline at end of file +% \input{content} +\end{document} From 5c2952787bcbded4ae47743e3d2fbd48612b694e Mon Sep 17 00:00:00 2001 From: jlh Date: Fri, 23 Aug 2024 17:05:22 -0400 Subject: [PATCH 4/6] add refs --- blueprint/print/print.pdf | Bin 140377 -> 141866 bytes blueprint/src/natural_model.tex | 45 ++++++++++++++++++++++++++++---- blueprint/src/refs.bib | 10 +++++++ blueprint/src/web.bbl | 4 +++ 4 files changed, 54 insertions(+), 5 deletions(-) diff --git a/blueprint/print/print.pdf b/blueprint/print/print.pdf 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b/blueprint/src/natural_model.tex @@ -1,18 +1,33 @@ +In this section we describe the categorical semantics of +HoTT via Natural Models. +This will not be a detailed account of the syntax of HoTT, +but will be a detailed account of what is needed to interpret such syntax. + +\medskip + +\begin{notation*} + We will have two universe sizes - one small and one large. + We denote the category of small sets as $\set$ and the large sets as $\Set$. + For example, we could take the small sets $\set$ to be those in $\Set$ bounded in cardinality + by some inaccessible cardinal. +\end{notation*} + \subsection{Types} -Assume an inaccessible cardinal $\lambda$. Write $\Set$ for the category of all sets. Say that a set $A$ is $\lambda$-small if $|A| < \lambda$. Write $\Set_\lambda$ for the full -subcategory of $\Set$ spanned by $\lambda$-small sets. +% Assume an inaccessible cardinal $\lambda$. Write $\Set$ for the category of all sets. Say that a set $A$ is $\lambda$-small if $|A| < \lambda$. Write $\Set_\lambda$ for the full +% subcategory of $\Set$ spanned by $\lambda$-small sets. -Let $\mathbb{C}$ be a small category, i.e.~a category whose class of objects is a set and -whose hom-classes are sets. +Let $\catC$ be a small category, i.e.~a category whose class of objects is a $\set$ and +whose hom-classes are from $\set$. % We do not assume that $\mathbb{C}$ is $\lambda$-small for % the moment. - We write $\pshC$ for the category of presheaves over $\catC$, \[ \pshC \defeq [\catC^\op, \Set] \] +\medskip + % Because of the assumption of the existence of $\lambda$, $\pshC$ has additional structure. Let % \[ % \Term \to \Type @@ -22,6 +37,26 @@ \subsection{Types} % \Type(c) \defeq \{ A \co (\catC_{/c})^\op \to \Set_{\lambda} \} % \] +\begin{defn} + Following Awodey \cite{awodey2017naturalmodelshomotopytype}, + we say that a map $\tp : \Term \to \Type$ is presentable when + any fiber of a representable is representable. + In other words, given any $\Ga \in \catC$ and a map $A : \yo (\Ga) \to \Type$, + there is some representable $\Ga \cdot A \in \catC$ and maps $\disp{A} : \Ga \cdot A \to \Ga$ + and $\var_{A} : \yo (\Ga \cdot A) \to \Term$ forming a pullback + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXHlvIChcXEdhIFxcY2RvdCBBKSJdLFswLDEsIlxceW8gKFxcR2EpIl0sWzEsMSwiXFxUeXBlIl0sWzEsMCwiXFxUZXJtIl0sWzAsMSwiXFx5byAoXFxkaXNwe0F9KSIsMl0sWzEsMiwiQSIsMl0sWzMsMl0sWzAsMywiXFx2YXJfQSJdXQ== +\[\begin{tikzcd} + {\yo (\Ga \cdot A)} & \Term \\ + {\yo (\Ga)} & \Type + \arrow["{\var_A}", from=1-1, to=1-2] + \arrow["{\yo (\disp{A})}"', from=1-1, to=2-1] + \arrow[from=1-2, to=2-2] + \arrow["A"', from=2-1, to=2-2] +\end{tikzcd}\] +\end{defn} + +\medskip + The Natural Model associated to a presentable map $\tp \co \Term \to \Type$ consists of \begin{itemize} \item contexts as objects $\Gamma, \Delta, \ldots \in \catC$, diff --git a/blueprint/src/refs.bib b/blueprint/src/refs.bib index 84b0cfe..d4386bb 100644 --- a/blueprint/src/refs.bib +++ b/blueprint/src/refs.bib @@ -28,3 +28,13 @@ @misc{joyalnlabmodelstructuresoncat title = {Model structures on Cat}, howpublished = {\url{https://ncatlab.org/joyalscatlab/published/Model+structures+on+Cat}}, } + +@misc{awodey2017naturalmodelshomotopytype, + title={Natural models of homotopy type theory}, + author={Steve Awodey}, + year={2017}, + eprint={1406.3219}, + archivePrefix={arXiv}, + primaryClass={math.CT}, + url={https://arxiv.org/abs/1406.3219}, +} diff --git a/blueprint/src/web.bbl b/blueprint/src/web.bbl index e67c156..ab8efc8 100644 --- a/blueprint/src/web.bbl +++ b/blueprint/src/web.bbl @@ -1,5 +1,9 @@ \begin{thebibliography}{Awo23} +\bibitem[Awo17]{awodey2017naturalmodelshomotopytype} +Steve Awodey. +\newblock Natural models of homotopy type theory, 2017. + \bibitem[Awo23]{awodey2023hofmannstreicheruniverses} Steve Awodey. \newblock On hofmann-streicher universes, 2023. 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+*.synctex.gz +*.out +*.log +*.fdb_latexmk +*.blg +*.bbl +*.fls +*.aux *.auctex-auto diff --git a/blueprint/src/groupoid_model.tex b/blueprint/src/groupoid_model.tex index 2456c49..7819410 100644 --- a/blueprint/src/groupoid_model.tex +++ b/blueprint/src/groupoid_model.tex @@ -14,9 +14,6 @@ \subsection{Classifying display maps} \medskip \begin{notation*} - We will have two universe sizes - one small and one large. - We denote the category of small sets as $\set$ and the large sets as $\Set$ - (in the previous sections this would have been $\Set_{\lambda}$ and $\Set$ respectively). We denote the category of small categories as $\cat$ and the large categories as $\Cat$. We denote the category of small groupoids as $\grpd$. %% and the large groupoids as $\GRPD$. @@ -567,98 +564,21 @@ \subsection{Groupoid fibrations} \medskip -\subsection{Polynomial endofunctors} +\subsection{Classifying type dependency} -\begin{defn}[Polynomial endofunctor on a morphism - in an locally Cartesian closed category] - Let $\catC$ be a locally Cartesian closed category - (we will take presheaves on small groupoids). - This means for each morphism $t : B \to A$ we have an adjoint triple - % https://q.uiver.app/#q=WzAsMyxbMCwyXSxbMSwyLCJcXGNhdEMgLyBBIl0sWzEsMCwiXFxjYXRDIC8gQiJdLFsyLDEsInRfKiIsMCx7Im9mZnNldCI6LTV9XSxbMiwxLCJ0XyEiLDIseyJvZmZzZXQiOjV9XSxbMSwyLCJ0XioiLDFdLFs1LDMsIiIsMix7ImxldmVsIjoxLCJzdHlsZSI6eyJuYW1lIjoiYWRqdW5jdGlvbiJ9fV0sWzQsNSwiIiwyLHsibGV2ZWwiOjEsInN0eWxlIjp7Im5hbWUiOiJhZGp1bmN0aW9uIn19XV0= - \[\begin{tikzcd} - & {\catC / B} \\ - \\ - {} & {\catC / A} - \arrow[""{name=0, anchor=center, inner sep=0}, "{t_*}", bend left, shift left=5, from=1-2, to=3-2] - \arrow[""{name=1, anchor=center, inner sep=0}, "{t_!}"', bend right, shift right=5, from=1-2, to=3-2] - \arrow[""{name=2, anchor=center, inner sep=0}, "{t^*}"{description}, from=3-2, to=1-2] - \arrow["\dashv"{anchor=center}, draw=none, from=1, to=2] - \arrow["\dashv"{anchor=center}, draw=none, from=2, to=0] - \end{tikzcd}\] - where $t^{*}$ is pullback, and $t_{!}$ is composition with $t$. - - Let $t : B \to A$ be a morphism in $\catC$. - Then define $\Poly{t} : \catC \to \catC$ be the composition - \[ - \Poly{t} := A_{!} \circ t_{*} \circ B^{*} - \quad \quad \quad - % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGNhdEMiXSxbMSwwLCJcXGNhdEMgLyBCIl0sWzIsMCwiXFxjYXRDIC8gQSJdLFszLDAsIlxcY2F0QyJdLFswLDEsIkJeKiJdLFsxLDIsInRfKiJdLFsyLDMsIkFfISJdXQ== - \begin{tikzcd} - \catC & {\catC / B} & {\catC / A} & \catC - \arrow["{B^*}", from=1-1, to=1-2] - \arrow["{t_*}", from=1-2, to=1-3] - \arrow["{A_!}", from=1-3, to=1-4] - \end{tikzcd}\] -\end{defn} - -\medskip - -\begin{prop} - The data of a map into the polynomial applied to an object in $\catC$ - % https://q.uiver.app/#q=WzAsMixbMCwwLCJYIl0sWzEsMCwiXFxQb2x5e3R9IFkiXSxbMCwxXV0= - \[\begin{tikzcd} - X & {\Poly{t} Y} - \arrow["\phi", from=1-1, to=1-2] - \end{tikzcd}\] - corresponds to - % https://q.uiver.app/#q=WzAsMyxbMCwwLCJYIl0sWzIsMCwiXFxQb2x5e3R9IFkiXSxbMSwxLCJBIl0sWzAsMSwiXFxwaGkiXSxbMSwyLCJ0XyogQl4qIFkiXSxbMCwyLCJcXGFsIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d - \[\begin{tikzcd} - X && {\Poly{t} Y} \\ - & A - \arrow["\phi", from=1-1, to=1-3] - \arrow["\al"', dashed, from=1-1, to=2-2] - \arrow["{t_* B^* Y}", from=1-3, to=2-2] - \end{tikzcd}\] - Applying the adjunction $A_{!} \dashv A^{*}$, this corresponds to - % https://q.uiver.app/#q=WzAsMyxbMCwwLCJCXyEgdF4qXFxhbCJdLFsyLDAsIkIgXFx0aW1lcyBZIl0sWzEsMSwiQiJdLFswLDEsIlxcdGlsZGV7XFxwaGl9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsMiwiQl4qIFkiXSxbMCwyLCJ0XipcXGFsIiwyXV0= - \[ - \al : X \to A - \quad \quad \text{ and } - \quad \quad - \begin{tikzcd} - {B_! t^*\al} && {B \times Y} \\ - & B - \arrow["{\tilde{\phi}}", dashed, from=1-1, to=1-3] - \arrow["{t^*\al}"', from=1-1, to=2-2] - \arrow["{B^* Y}", from=1-3, to=2-2] - \end{tikzcd}\] - Applying the adjunction $t^{*} \dashv t_{*}$, - this corresponds to - % https://q.uiver.app/#q=WzAsMixbMCwwLCJCXyEgdF4qXFxhbCJdLFsyLDAsIlkiXSxbMCwxLCJcXHRpbGRle1xcdGlsZGV7XFxwaGl9fSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ== - \[ - \al : X \to A - \quad \quad \text{ and } - \quad \quad - \begin{tikzcd} - {B_! t^*\al} && Y - \arrow["{\beta}", dashed, from=1-1, to=1-3] - \end{tikzcd}\] -\end{prop} - -\medskip - -\begin{prop}[$\Poly{\tp}\Type$ classifies dependent types]\label{tp_classifies} +\begin{prop}[$\Poly{\tp}$ classifies type dependency]\label{tp_classifies} Specialized to $\tp : \Term \to \Type$ in $\Pshgrpd$, - the previous proposition says that a map + the characterizing property of polynomial endofunctors + \cref{polynomial_endofunctor_property} says that a map from a representable - $\Ga \to \Poly{\tp}\Type$ + $\Ga \to \Poly{\tp}X$ corresponds to the data of \[ A : \Ga \to \Type \quad \quad \text{ and } \quad \quad - B : \Ga \cdot A \to \Type + B : \Ga \cdot A \to X \] which by Yoneda corresponds to the data in $\Cat$ of \[ @@ -666,21 +586,31 @@ \subsection{Polynomial endofunctors} \quad \quad \text{ and } \quad \quad - B : \Ga \cdot A \to \grpd + B : \Ga \cdot A \to X \] + The special case of when $X$ is also $\Type$ gives us a + classifier for dependent types. Furthermore, if $\si : \De \to \Ga$ were a representable map, - then we have a naturality square -% https://q.uiver.app/#q=WzAsNixbMywwLCJcXFBvbHl7XFx0cH0gW1xcR2EsXFxjYXRDXSAiXSxbMywxLCJcXFBvbHl7XFx0cH0gW1xcRGUsXFxjYXRDXSAiXSxbMCwxLCJcXERlIl0sWzAsMCwiXFxHYSJdLFsyLDAsIlxcU2lfe0EgXFxpbiBbXFxHYSxcXGdycGRdfSAgW1xcR2EuQSxcXGNhdENdIl0sWzIsMSwiXFxTaV97QSBcXGluIFtcXERlLFxcZ3JwZF19ICBbXFxEZS5BLFxcY2F0Q10iXSxbMiwzLCJcXHNpIl0sWzAsMSwiXFxQb2x5e1xcdHB9ey0gXFxjaXJjIFxcc2l9IiwxXSxbNCwwLCJcXGlzbyJdLFs1LDEsIlxcaXNvIiwyXSxbNCw1LCIoLSBcXGNpcmMgXFxzaSwtIFxcY2lyYyBcXHNpX0EpIiwxXV0= + then we have +% https://q.uiver.app/#q=WzAsNCxbMiwxLCJcXFBvbHl7XFx0cH0gWCJdLFswLDFdLFsxLDEsIlxcR2EiXSxbMSwwLCJcXERlIl0sWzMsMiwiXFxzaSIsMl0sWzMsMCwiKEFcXGNpcmMgXFxzaSwgQiBcXGNpcmMgXFx0cF4qIFxcc2kpIl0sWzIsMCwiKEEsIEIpIiwyXV0= \[\begin{tikzcd} - \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\catC]} & {\Poly{\tp} \Type \, \Ga } \\ - \De && {\Si_{A \in [\De,\grpd]} [\De.A,\catC]} & {\Poly{\tp} \Type \, \De } - \arrow["\iso", from=1-3, to=1-4] - \arrow["{(- \circ \si,- \circ \si_A)}"{description}, from=1-3, to=2-3] - \arrow["{\Poly{\tp}{- \circ \si}}"{description}, from=1-4, to=2-4] - \arrow["\si", from=2-1, to=1-1] - \arrow["\iso"', from=2-3, to=2-4] + & \De \\ + {} & \Ga & {\Poly{\tp} X} + \arrow["\si"', from=1-2, to=2-2] + \arrow["{(A\circ \si, B \circ \tp^* \si)}", from=1-2, to=2-3] + \arrow["{(A, B)}"', from=2-2, to=2-3] \end{tikzcd}\] +% % https://q.uiver.app/#q=WzAsNixbMywwLCJcXFBvbHl7XFx0cH0gW1xcR2EsXFxjYXRDXSAiXSxbMywxLCJcXFBvbHl7XFx0cH0gW1xcRGUsXFxjYXRDXSAiXSxbMCwxLCJcXERlIl0sWzAsMCwiXFxHYSJdLFsyLDAsIlxcU2lfe0EgXFxpbiBbXFxHYSxcXGdycGRdfSAgW1xcR2EuQSxcXGNhdENdIl0sWzIsMSwiXFxTaV97QSBcXGluIFtcXERlLFxcZ3JwZF19ICBbXFxEZS5BLFxcY2F0Q10iXSxbMiwzLCJcXHNpIl0sWzAsMSwiXFxQb2x5e1xcdHB9ey0gXFxjaXJjIFxcc2l9IiwxXSxbNCwwLCJcXGlzbyJdLFs1LDEsIlxcaXNvIiwyXSxbNCw1LCIoLSBcXGNpcmMgXFxzaSwtIFxcY2lyYyBcXHNpX0EpIiwxXV0= +% \[\begin{tikzcd} +% \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\catC]} & {\Poly{\tp} \Type \, \Ga } \\ +% \De && {\Si_{A \in [\De,\grpd]} [\De.A,\catC]} & {\Poly{\tp} \Type \, \De } +% \arrow["\iso", from=1-3, to=1-4] +% \arrow["{(- \circ \si,- \circ \si_A)}"{description}, from=1-3, to=2-3] +% \arrow["{\Poly{\tp}{- \circ \si}}"{description}, from=1-4, to=2-4] +% \arrow["\si", from=2-1, to=1-1] +% \arrow["\iso"', from=2-3, to=2-4] +% \end{tikzcd}\] \end{prop} \medskip diff --git a/blueprint/src/polynomial.tex b/blueprint/src/polynomial.tex new file mode 100644 index 0000000..57e4124 --- /dev/null +++ b/blueprint/src/polynomial.tex @@ -0,0 +1,97 @@ +In this section we develop some of the definitions +and lemmas related to polynomial endofunctors that we will use +in the rest of the notes. + +\begin{defn}[Polynomial endofunctor] + Let $\catC$ be a locally Cartesian closed category + (in our case, presheaves on the category of contexts). + This means for each morphism $t : B \to A$ we have an adjoint triple + % https://q.uiver.app/#q=WzAsMyxbMCwyXSxbMSwyLCJcXGNhdEMgLyBBIl0sWzEsMCwiXFxjYXRDIC8gQiJdLFsyLDEsInRfKiIsMCx7Im9mZnNldCI6LTV9XSxbMiwxLCJ0XyEiLDIseyJvZmZzZXQiOjV9XSxbMSwyLCJ0XioiLDFdLFs1LDMsIiIsMix7ImxldmVsIjoxLCJzdHlsZSI6eyJuYW1lIjoiYWRqdW5jdGlvbiJ9fV0sWzQsNSwiIiwyLHsibGV2ZWwiOjEsInN0eWxlIjp7Im5hbWUiOiJhZGp1bmN0aW9uIn19XV0= + \[\begin{tikzcd} + & {\catC / B} \\ + \\ + {} & {\catC / A} + \arrow[""{name=0, anchor=center, inner sep=0}, "{t_*}", bend left, shift left=5, from=1-2, to=3-2] + \arrow[""{name=1, anchor=center, inner sep=0}, "{t_!}"', bend right, shift right=5, from=1-2, to=3-2] + \arrow[""{name=2, anchor=center, inner sep=0}, "{t^*}"{description}, from=3-2, to=1-2] + \arrow["\dashv"{anchor=center}, draw=none, from=1, to=2] + \arrow["\dashv"{anchor=center}, draw=none, from=2, to=0] + \end{tikzcd}\] + where $t^{*}$ is pullback, and $t_{!}$ is composition with $t$. + + Let $t : B \to A$ be a morphism in $\catC$. + Then define $\Poly{t} : \catC \to \catC$ be the composition + \[ + \Poly{t} := A_{!} \circ t_{*} \circ B^{*} + \quad \quad \quad + % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGNhdEMiXSxbMSwwLCJcXGNhdEMgLyBCIl0sWzIsMCwiXFxjYXRDIC8gQSJdLFszLDAsIlxcY2F0QyJdLFswLDEsIkJeKiJdLFsxLDIsInRfKiJdLFsyLDMsIkFfISJdXQ== + \begin{tikzcd} + \catC & {\catC / B} & {\catC / A} & \catC + \arrow["{B^*}", from=1-1, to=1-2] + \arrow["{t_*}", from=1-2, to=1-3] + \arrow["{A_!}", from=1-3, to=1-4] + \end{tikzcd}\] +\end{defn} + +\medskip + +\begin{prop}[Characterising property of Polynomial Endofunctors] + \label{polynomial_endofunctor_property} + The data of a map into the polynomial applied to an object in $\catC$ +% https://q.uiver.app/#q=WzAsMixbMCwwLCJcXEdhIl0sWzEsMCwiXFxQb2x5e3R9IFkiXSxbMCwxXV0= +\[\begin{tikzcd} + \Ga & {\Poly{t} Y} + \arrow[from=1-1, to=1-2] +\end{tikzcd}\] + corresponds to +% https://q.uiver.app/#q=WzAsMyxbMCwwLCJcXEdhIl0sWzIsMCwiXFxQb2x5e3R9IFkiXSxbMSwxLCJBIl0sWzAsMSwiXFxwaGkiXSxbMSwyLCJ0XyogQl4qIFkiXSxbMCwyLCJcXGFsIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d +\[\begin{tikzcd} + \Ga && {\Poly{t} Y} \\ + & A + \arrow["\phi", from=1-1, to=1-3] + \arrow["\al"', dashed, from=1-1, to=2-2] + \arrow["{t_* B^* Y}", from=1-3, to=2-2] +\end{tikzcd}\] + Applying the adjunction $A_{!} \dashv A^{*}$, this corresponds to + \[ + \al : \Ga \to A + \quad \quad \text{ and } + \quad \quad + % https://q.uiver.app/#q=WzAsMyxbMCwwLCJCXyEgdF4qXFxhbCJdLFsyLDAsIkIgXFx0aW1lcyBZIl0sWzEsMSwiQiJdLFswLDEsIlxcdGlsZGV7XFxwaGl9IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzEsMiwiQl4qIFkiXSxbMCwyLCJ0XipcXGFsIiwyXV0= + \begin{tikzcd} + {B_! t^*\al} && {B \times Y} \\ + & B + \arrow["{\tilde{\phi}}", dashed, from=1-1, to=1-3] + \arrow["{t^*\al}"', from=1-1, to=2-2] + \arrow["{B^* Y}", from=1-3, to=2-2] + \end{tikzcd} +\] + Applying the adjunction $t^{*} \dashv t_{*}$, + this corresponds to + % https://q.uiver.app/#q=WzAsMixbMCwwLCJCXyEgdF4qXFxhbCJdLFsyLDAsIlkiXSxbMCwxLCJcXHRpbGRle1xcdGlsZGV7XFxwaGl9fSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ== + \[ + \al : \Ga \to A + \quad \quad \text{ and } + \quad \quad + \begin{tikzcd} + {B_! t^*\al} && Y + \arrow["{\beta}", dashed, from=1-1, to=1-3] + \end{tikzcd}\] + + Henceforth we will write + \[ (\al, \be) : \Ga \to \Poly{t} Y\] + for this map, + since it is uniquely determined by this data. + We also have that for any $\si : \De \to \Ga$, + the following triangle commutes +% https://q.uiver.app/#q=WzAsNCxbMiwxLCJcXFBvbHl7dH0gWSJdLFswLDFdLFsxLDEsIlxcR2EiXSxbMSwwLCJcXERlIl0sWzMsMiwiXFxzaSIsMl0sWzMsMCwiKFxcYWwgXFxjaXJjIFxcc2ksIFxcYmUgXFxjaXJjIHReKiBcXHNpKSJdLFsyLDAsIihcXGFsLCBcXGJlKSIsMl1d +\[\begin{tikzcd} + & \De \\ + {} & \Ga & {\Poly{t} Y} + \arrow["\si"', from=1-2, to=2-2] + \arrow["{(\al \circ \si, \be \circ t^* \si)}", from=1-2, to=2-3] + \arrow["{(\al, \be)}"', from=2-2, to=2-3] +\end{tikzcd}\] +\end{prop} + +\medskip diff --git a/blueprint/src/print.tex b/blueprint/src/print.tex index b10d7e8..b10e339 100644 --- a/blueprint/src/print.tex +++ b/blueprint/src/print.tex @@ -41,13 +41,15 @@ \begin{document} \maketitle - \section{Natural Models} \input{natural_model} \section{The Groupoid Model} \input{groupoid_model} +\section{Polynomial Endofunctors} +\input{polynomial} + \bibliography{refs.bib}{} \bibliographystyle{alpha} From b734dd92fb481ba1583104c28a3d31ecdc79c0a5 Mon Sep 17 00:00:00 2001 From: jlh Date: Fri, 23 Aug 2024 18:26:26 -0400 Subject: [PATCH 6/6] refactor polynomial section --- blueprint/print/print.pdf | Bin 143686 -> 143649 bytes blueprint/src/groupoid_model.tex | 86 ++++++++++++++----------------- 2 files changed, 40 insertions(+), 46 deletions(-) diff --git a/blueprint/print/print.pdf b/blueprint/print/print.pdf index f17f41ed8443f444a4e99ac6db0351e8249227ad..f02f011724b6c0b8c734fc600e964dbfdc8fd459 100644 GIT binary patch delta 22712 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zEm6kV($yxTH!`&SAk6;6aQ5~nI$sD~q!EvBp#PQ(v_(7)FP6kV+{AWGsA~{gKzKbGmP>ShKU%p)KQ_n{_PxtamXag|hIA!q>2|2; zA-$Whr5Bt#?%?#(pKeZL?^F)TT#L9jYxB-cY%ok{^_F^_UGl_lM0~rA^onmtZr)odVXPiIQjl4G{&QWg^~>&1 zcd*I9y!D}gI;_{2iLd*>fzOESw^!Oi z3vmxJEB7wv1B@nC7XQr3SHNGO2DM+Fht;m;ryxnj?)>Whn#BHQE*JDx5RC~}LQD{s Lon1v+759GtcSpM> diff --git a/blueprint/src/groupoid_model.tex b/blueprint/src/groupoid_model.tex index 7819410..dc65a9f 100644 --- a/blueprint/src/groupoid_model.tex +++ b/blueprint/src/groupoid_model.tex @@ -601,16 +601,20 @@ \subsection{Classifying type dependency} \arrow["{(A\circ \si, B \circ \tp^* \si)}", from=1-2, to=2-3] \arrow["{(A, B)}"', from=2-2, to=2-3] \end{tikzcd}\] -% % https://q.uiver.app/#q=WzAsNixbMywwLCJcXFBvbHl7XFx0cH0gW1xcR2EsXFxjYXRDXSAiXSxbMywxLCJcXFBvbHl7XFx0cH0gW1xcRGUsXFxjYXRDXSAiXSxbMCwxLCJcXERlIl0sWzAsMCwiXFxHYSJdLFsyLDAsIlxcU2lfe0EgXFxpbiBbXFxHYSxcXGdycGRdfSAgW1xcR2EuQSxcXGNhdENdIl0sWzIsMSwiXFxTaV97QSBcXGluIFtcXERlLFxcZ3JwZF19ICBbXFxEZS5BLFxcY2F0Q10iXSxbMiwzLCJcXHNpIl0sWzAsMSwiXFxQb2x5e1xcdHB9ey0gXFxjaXJjIFxcc2l9IiwxXSxbNCwwLCJcXGlzbyJdLFs1LDEsIlxcaXNvIiwyXSxbNCw1LCIoLSBcXGNpcmMgXFxzaSwtIFxcY2lyYyBcXHNpX0EpIiwxXV0= -% \[\begin{tikzcd} -% \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\catC]} & {\Poly{\tp} \Type \, \Ga } \\ -% \De && {\Si_{A \in [\De,\grpd]} [\De.A,\catC]} & {\Poly{\tp} \Type \, \De } -% \arrow["\iso", from=1-3, to=1-4] -% \arrow["{(- \circ \si,- \circ \si_A)}"{description}, from=1-3, to=2-3] -% \arrow["{\Poly{\tp}{- \circ \si}}"{description}, from=1-4, to=2-4] -% \arrow["\si", from=2-1, to=1-1] -% \arrow["\iso"', from=2-3, to=2-4] -% \end{tikzcd}\] + where $\tp^{*} \si$ is given by +% https://q.uiver.app/#q=WzAsNixbMSwwLCJcXEdhXFxjZG90IEEiXSxbMSwxLCJcXEdhIl0sWzIsMSwiXFxncnBkIl0sWzIsMCwiXFxwdGdycGQiXSxbMCwxLCJcXERlIl0sWzAsMCwiXFxEZSBcXGNkb3QgQVxcY2lyYyBcXHNpIl0sWzAsMSwiIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzEsMiwiQSIsMl0sWzMsMl0sWzAsM10sWzQsMSwiXFxzaSIsMl0sWzUsNCwiIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzUsMCwiXFx0cF4qIFxcc2kiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= +\[\begin{tikzcd} + {\De \cdot A\circ \si} & {\Ga\cdot A} & \ptgrpd \\ + \De & \Ga & \grpd + \arrow["{\tp^* \si}", dashed, from=1-1, to=1-2] + \arrow[two heads, from=1-1, to=2-1] + \arrow[from=1-2, to=1-3] + \arrow[two heads, from=1-2, to=2-2] + \arrow[from=1-3, to=2-3] + \arrow["\si"', from=2-1, to=2-2] + \arrow["A"', from=2-2, to=2-3] +\end{tikzcd}\] + \end{prop} \medskip @@ -620,49 +624,39 @@ \subsection{Pi and Sigma structure} %% TODO fix subsection title \medskip \begin{lemma}\label{BC_iff_nat_trans} - Let $\catC$ be a large category, - and let $[-,\catC] \in \Pshgrpd$ be the restriction of the Yoneda embedding - $\yo : \Cat \to \PshCat$. + $X \in \Pshgrpd$ be a presheaf. Let $F$ be an operation that takes a groupoid $\Ga$, a functor $A : \Ga \to \grpd$ - and $B : \Ga \cdot A \to \catC$ and returns a functor - $F_{A} B : \Ga \to \catC$. + and $B : \Ga \cdot A \to X$ and returns a natural transformation + $F_{A} B : \Ga \to X$. - Then $\tilde{F} : \Poly{\tp} [-,\catC] \to [-,\catC]$ - \[ \tilde{F}_{\Ga} (A,B) = F_{A}B \] - defines a natural transformation + Then using Yoneda to define $\tilde{F} : \Poly{\tp} X \to X$ pointwise as + \begin{align*} + \tilde{F}_{\Ga} : \Pshgrpd(\Ga, \Poly{\tp} X) & \to \Pshgrpd(\Ga, X) \\ + (A, B) & \mapsto F_{A} B + \end{align*} + gives us a natural transformation if and only if $F$ satisfies the strict Beck-Chevalley condition - \[ (F_{A} B) \circ \si = F_{A \circ \si} (B \circ \si_{A})\] - where $\si_{A}$ is given by - % https://q.uiver.app/#q=WzAsNixbMSwwLCJcXEdhXFxjZG90IEEiXSxbMSwxLCJcXEdhIl0sWzIsMSwiXFxncnBkIl0sWzIsMCwiXFxwdGdycGQiXSxbMCwxLCJcXERlIl0sWzAsMCwiXFxEZSBcXGNkb3QgQVxcY2lyYyBcXHNpIl0sWzAsMSwiIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzEsMiwiQSIsMl0sWzMsMl0sWzAsM10sWzQsMSwiXFxzaSIsMl0sWzUsNCwiIiwyLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzUsMCwiXFxzaV9BIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d -\[\begin{tikzcd} - {\De \cdot A\circ \si} & {\Ga\cdot A} & \ptgrpd \\ - \De & \Ga & \grpd - \arrow["{\si_A}", dashed, from=1-1, to=1-2] - \arrow[two heads, from=1-1, to=2-1] - \arrow[from=1-2, to=1-3] - \arrow[two heads, from=1-2, to=2-2] - \arrow[from=1-3, to=2-3] - \arrow["\si"', from=2-1, to=2-2] - \arrow["A"', from=2-2, to=2-3] -\end{tikzcd}\] + \[ (F_{A} B) \circ \si = F_{A \circ \si} (B \circ \tp^{*} \si)\] + for every $\si : \De \to \Ga$ in $\grpd$. \end{lemma} \begin{proof} Using \cref{tp_classifies} -% https://q.uiver.app/#q=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 -\[\begin{tikzcd} - \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\catC]} & {\Poly{\tp} [\Ga,\catC] } & {[\Ga,\catC]} \\ - \De && {\Si_{A \in [\De,\grpd]} [\De.A,\catC]} & {\Poly{\tp} [\De,\catC] } & {[\De,\catC] } - \arrow[from=1-3, to=1-4] - \arrow["F"{description}, bend left, from=1-3, to=1-5] - \arrow["{(- \circ \si,- \circ \si_A)}"{description}, from=1-3, to=2-3] - \arrow["{F_\Ga}"', from=1-4, to=1-5] - \arrow["{\Poly{\tp}{- \circ \si}}"', from=1-4, to=2-4] - \arrow["{- \circ \si}", from=1-5, to=2-5] - \arrow["\si", from=2-1, to=1-1] - \arrow[from=2-3, to=2-4] - \arrow["F"{description}, bend right, from=2-3, to=2-5] - \arrow["{F_\De}", from=2-4, to=2-5] +% https://q.uiver.app/#q=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 +\[\begin{tikzcd}[column sep = small] + {(A, B)} &&& {F_AB} \\ + & {\Pshgrpd(\Ga, \Poly{\tp} X) } & {\Pshgrpd(\Ga, X)} \\ + & {\Pshgrpd(\De, \Poly{\tp} X) } & {\Pshgrpd(\De, X)} \\ + {(A \circ \si, B \circ \tp^* \si)} && {F_{A \circ \si} B \circ \tp^* \si} & {(F_A B) \circ \si} + \arrow[maps to, from=1-1, to=1-4] + \arrow[maps to, from=1-1, to=4-1] + \arrow[maps to, from=1-4, to=4-4] + \arrow["{\tilde{F}_{\Ga}}", from=2-2, to=2-3] + \arrow["{- \circ \si}"', from=2-2, to=3-2] + \arrow["{- \circ \si}", from=2-3, to=3-3] + \arrow["{\tilde{F}_{\De}}"', from=3-2, to=3-3] + \arrow[maps to, from=4-1, to=4-3] + \arrow[Rightarrow, dashed, no head, from=4-3, to=4-4] \end{tikzcd}\] \end{proof}