From 0b0945ae076afad850886dd0ba52643d254cb69a Mon Sep 17 00:00:00 2001 From: jlh Date: Wed, 7 Aug 2024 11:13:20 -0400 Subject: [PATCH] natural model pi --- Notes/natural_model_univ/groupoid_model.tex | 126 +++++++++++++++----- Notes/natural_model_univ/main.pdf | Bin 313817 -> 319977 bytes Notes/natural_model_univ/main.tex | 1 + 3 files changed, 96 insertions(+), 31 deletions(-) diff --git a/Notes/natural_model_univ/groupoid_model.tex b/Notes/natural_model_univ/groupoid_model.tex index 9967c3b..d050b58 100644 --- a/Notes/natural_model_univ/groupoid_model.tex +++ b/Notes/natural_model_univ/groupoid_model.tex @@ -430,7 +430,7 @@ \subsection{Groupoid fibrations} \medskip -\begin{defn}[$\Pi$-former operation] +\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 @@ -648,7 +648,7 @@ \subsection{Polynomial endofunctors} \medskip -\begin{prop}[$\Poly{\tp}\Type$ classifies dependent types] +\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 @@ -670,6 +670,18 @@ \subsection{Polynomial endofunctors} 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= +\[\begin{tikzcd} + \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\catC]} & {\Poly{\tp} [\Ga,\catC] } \\ + \De && {\Si_{A \in [\De,\grpd]} [\De.A,\catC]} & {\Poly{\tp} [\De,\catC] } + \arrow["\iso", from=1-3, to=1-4] + \arrow["{(- \circ \si,- \circ \si_A)}"{description}, from=1-3, to=2-3] + \arrow["{\Poly{\tp}{- \circ \si}}"{description}, from=1-4, to=2-4] + \arrow["\si", from=2-1, to=1-1] + \arrow["\iso"', from=2-3, to=2-4] +\end{tikzcd}\] \end{prop} \medskip @@ -678,26 +690,40 @@ \subsection{$\Pi$ and $\Si$ structure} \medskip -\begin{lemma} +\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 + 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 $F : \Poly{\tp} [-,\catC] \to [-,\catC]$ - \[ F_{\Ga} (A,B) = F_{A}B \] + 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 strict the Beck-Chevalley condition + 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} -% https://q.uiver.app/#q=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 + Using \cref{tp_classifies} +% https://q.uiver.app/#q=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 \[\begin{tikzcd} - \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A \to \catC]} & {\Poly{\tp} [\Ga,\catC] } & {[\Ga,\catC]} \\ - \De && {\Si_{A \in [\De,\grpd]} [\De.A \to \catC]} & {\Poly{\tp} [\De,\catC] } & {[\De,\catC] } + \Ga && {\Si_{A \in [\Ga,\grpd]} [\Ga.A,\catC]} & {\Poly{\tp} [\Ga,\catC] } & {[\Ga,\catC]} \\ + \De && {\Si_{A \in [\De,\grpd]} [\De.A,\catC]} & {\Poly{\tp} [\De,\catC] } & {[\De,\catC] } \arrow[from=1-3, to=1-4] \arrow["F"{description}, bend left, from=1-3, to=1-5] \arrow["{(- \circ \si,- \circ \si_A)}"{description}, from=1-3, to=2-3] @@ -711,32 +737,70 @@ \subsection{$\Pi$ and $\Si$ structure} \end{tikzcd}\] \end{proof} -\begin{defn}[Interpretation of $\Pi$ and $\la$] +\begin{defn}[Interpretation of $\Pi$ types] We define the natural transformation $\Pi : \Poly{\tp} \Type \to \Type$ - by first taking some small groupoid $\Ga$ and defining - \[\Pi_{\Ga} : \Pshgrpd(\Ga, \Poly{\tp} \Type) \to \Pshgrpd(\Ga, \Type)\] - Let $(A,B) : \Ga \to \Poly{\tp} \Type$, corresponding to $A : \Ga \to \grpd$ - and $B : \Ga \cdot A \to \grpd$. - Taking the pushforward of fibrations in $\grpd$ - (formally defined as operations on the classifying maps), - we obtain $\Pi_{A}B : \Ga \to \grpd$ corresponding by Yoneda - to an element of $\Pshgrpd(\Ga, \Type)$. - - As indicated in the diagram, we take this to be the pushforward of the - dependent display map $\disp{B}$ along the display map it depends on $\disp{A}$. - Note that this pushforward is in $\grpd$, - and this pushforward is only defined on fibrations. - - TODO: define $\la$. + 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} - TODO: naturality. - % Beck-chevalley. - % Steve says the coherence problem is not an issue here - - % i.e. that our equalities are strict. + 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 +\[\begin{tikzcd} + {\Poly{\tp}{\Term} \, \Ga} & {[\Ga,\ptgrpd]} \\ + {\Poly{\tp}{\Type} \, \Ga} & {[\Ga,\grpd]} + \arrow["{\la_\Ga}", from=1-1, to=1-2] + \arrow["{\Poly{\tp}{\tp}_\Ga}"', from=1-1, to=2-1] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] + \arrow["{U \circ -}", from=1-2, to=2-2] + \arrow["{\Pi_\Ga}"', from=2-1, to=2-2] +\end{tikzcd}\] - TODO: prove pullback. +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 diff --git a/Notes/natural_model_univ/main.pdf 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