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a/blueprint/src/groupoid_model.tex b/blueprint/src/groupoid_model.tex index 686fa34..0ea01f8 100644 --- a/blueprint/src/groupoid_model.tex +++ b/blueprint/src/groupoid_model.tex @@ -545,25 +545,73 @@ \subsection{Groupoid fibrations} \medskip \begin{prop}[Path object fibration] - Let $\Ga$ be a small groupoid. + Let $A$ be a small groupoid. Recall that $\grpd$ is Cartesian closed, - so we can take the image of the above diagram under the functor $\Ga^{-}$. + so we can take the image of the above diagram under the functor $A^{-}$. % https://q.uiver.app/#q=WzAsNCxbMSwxLCJcXEdhIFxcdGltZXMgXFxHYSJdLFsxLDIsIlxcR2EiXSxbMiwxLCJcXEdhIl0sWzAsMCwiXFxHYV5cXEludGVydmFsIl0sWzAsMV0sWzAsMl0sWzMsMSwiXFxHYV57XFxkZV8wfSIsMix7ImN1cnZlIjoxLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywyLCJcXEdhXntcXGRlXzF9IiwwLHsiY3VydmUiOi0xLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMywwLCJcXEdhXlxccGFydGlhbCIsMSx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9LCJoZWFkIjp7Im5hbWUiOiJlcGkifX19XV0= \[\begin{tikzcd} - {\Ga^{\Interval}} \\ - & {\Ga \times \Ga} & \Ga \\ - & \Ga - \arrow["{\Ga^\partial}"{description}, dashed, two heads, from=1-1, to=2-2] - \arrow["{\Ga^{\de_1}}", bend left, two heads, from=1-1, to=2-3] - \arrow["{\Ga^{\de_0}}"', bend right, two heads, from=1-1, to=3-2] + {A^{\Interval}} \\ + & {A \times A} & A \\ + & A + \arrow["{A^\partial}"{description}, dashed, two heads, from=1-1, to=2-2] + \arrow["{A^{\de_1}}", bend left, two heads, from=1-1, to=2-3] + \arrow["{A^{\de_0}}"', bend right, two heads, from=1-1, to=3-2] \arrow[from=2-2, to=2-3] \arrow[from=2-2, to=3-2] \end{tikzcd}\] Then the indicated morphisms are fibrations, - and $\Ga^{\de_{0}}, \Ga^{\de_{1}}$ form adjoint equivalences - with $\Ga^{!} : \Ga \to \Ga^{\Interval}$. + and $A^{\de_{0}}, A^{\de_{1}}$ form adjoint equivalences + with $A^{!} : A \to A^{\Interval}$. \end{prop} +We can use this to justify the interpretation of the identity type later, +where we will have the strictified versions (as in strictly stable under substitution) +of the above +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + A & {\terminal \cdot A} & \ptgrpd & \ptgrpd \\ + {A^{\Interval}} & {\terminal \cdot A \cdot A \cdot \Id} & {I'} & \grpd \\ + {A \times A} & {\terminal \cdot A \cdot A} & {U \times_\grpd U} & \ptgrpd \\ + A & {\terminal \cdot A} & \ptgrpd & \grpd \\ + {} & \terminal & \grpd + \arrow["\iso"{description}, from=1-1, to=1-2] + \arrow[from=1-1, to=2-1] + \arrow["{\var_A}", from=1-2, to=1-3] + \arrow["{A^* \rho'}"{description}, from=1-2, to=2-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-2, to=2-3] + \arrow["{\rho'}"{description}, from=1-3, to=2-3] + \arrow["U", from=1-4, to=2-4] + \arrow["\iso"{description}, from=2-1, to=2-2] + \arrow["{A^{\partial}}"{description}, two heads, from=2-1, to=3-1] + \arrow[from=2-2, to=2-3] + \arrow["{\disp{\Id' \circ U^* \var_A}}"{description}, from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow[from=2-3, to=1-4] + \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=30}, draw=none, from=2-3, to=2-4] + \arrow[from=2-3, to=3-3] + \arrow["\iso"{description}, from=3-1, to=3-2] + \arrow["\fst"', from=3-1, to=4-1] + \arrow[from=3-2, to=3-3] + \arrow["{\disp{U \circ \var_A}}"{description}, from=3-2, to=4-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-2, to=4-3] + \arrow["{\Id'}"{description}, from=3-3, to=2-4] + \arrow["\snd"', from=3-3, to=3-4] + \arrow["\fst"{description}, from=3-3, to=4-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-3, to=4-4] + \arrow["U"', from=3-4, to=4-4] + \arrow["\iso"{description}, from=4-1, to=4-2] + \arrow["{\var_A}"{description}, from=4-2, to=4-3] + \arrow["{\disp{A}}"', from=4-2, to=5-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=4-2, to=5-3] + \arrow["U"', from=4-3, to=4-4] + \arrow["U", from=4-3, to=5-3] + \arrow["A"', from=5-2, to=5-3] +\end{tikzcd}\] + +In general, we will want to build a pathspace for a type in any context, +which requires us to pull back the interval along the context, +and rebuild the required fibration by exponentiation in the slice. + \medskip \subsection{Classifying type dependency} @@ -1210,31 +1258,344 @@ \subsection{Identity types} in \cref{id_diagram1} commutes. \end{proof} -% \begin{lemma} -% We can then construct the pullback $I'$ -% % https://q.uiver.app/#q=WzAsNSxbMCwwLCJcXHB0Z3JwZCJdLFsxLDIsIlUgXFx0aW1lc19cXGdycGQgVSJdLFsyLDIsIlxcZ3JwZCJdLFsyLDEsIlxccHRncnBkIl0sWzEsMSwiSSciXSxbMCwxLCJcXGRlIiwyLHsiY3VydmUiOjJ9XSxbMSwyLCJcXElkJyIsMl0sWzMsMiwiVSJdLFswLDMsIlxccmVmbCciLDAseyJjdXJ2ZSI6LTJ9XSxbNCwxXSxbNCwzXSxbMCw0LCJcXHJobyciLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNCwyLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0= -% \[\begin{tikzcd} -% \ptgrpd \\ -% & {I'} & \ptgrpd \\ -% & {U \times_\grpd U} & \grpd -% \arrow["{\rho'}"{description}, dashed, from=1-1, to=2-2] -% \arrow["{\refl'}", bend left, from=1-1, to=2-3] -% \arrow["\de"', bend right, from=1-1, to=3-2] -% \arrow[from=2-2, to=2-3] -% \arrow[from=2-2, to=3-2] -% \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] -% \arrow["U", from=2-3, to=3-3] -% \arrow["{\Id'}"', from=3-2, to=3-3] -% \end{tikzcd}\] -% as the groupoid with objects $(A, a_{0}, a_{1}, h)$ -% where $A$ is a groupoid with $a_{0}, a_{1} \in A$ and $h : a_{0} \to a_{1}$, -% and morphisms -% \[(f, \phi_{0}, \phi_{1}, A h = k) : -% (A, a_{0}, a_{1}, h : a_{0} \to a_{1}) \to -% (B, b_{0}, b_{1}, k : b_{0} \to b_{1})\] -% where $f : A \to B$, -% $\phi_{i} : f a_{i} \to b_{i}$ and $A h = k$ represents a merely propositional -% proof of equality. -% Then we can also compute -% \[\rho' (A, a) = (A, a, a, \id_{a})\] -% \end{lemma} +\begin{lemma} + \label{I'_classifies} + We can then construct the pullback $I'$ + % https://q.uiver.app/#q=WzAsNSxbMCwwLCJcXHB0Z3JwZCJdLFsxLDIsIlUgXFx0aW1lc19cXGdycGQgVSJdLFsyLDIsIlxcZ3JwZCJdLFsyLDEsIlxccHRncnBkIl0sWzEsMSwiSSciXSxbMCwxLCJcXGRlIiwyLHsiY3VydmUiOjJ9XSxbMSwyLCJcXElkJyIsMl0sWzMsMiwiVSJdLFswLDMsIlxccmVmbCciLDAseyJjdXJ2ZSI6LTJ9XSxbNCwxXSxbNCwzXSxbMCw0LCJcXHJobyciLDEseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNCwyLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0= +\[\begin{tikzcd} + \ptgrpd \\ + & {I'} & \ptgrpd \\ + & {U \times_\grpd U} & \grpd + \arrow["{\rho'}"{description}, dashed, from=1-1, to=2-2] + \arrow["{\refl'}", bend left, from=1-1, to=2-3] + \arrow["\de"', bend right, from=1-1, to=3-2] + \arrow[from=2-2, to=2-3] + \arrow[from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["U", from=2-3, to=3-3] + \arrow["{\Id'}"', from=3-2, to=3-3] +\end{tikzcd}\] + as the groupoid with objects $(A, a_{0}, a_{1}, h)$ + where $A$ is a groupoid with $a_{0}, a_{1} \in A$ and $h : a_{0} \to a_{1}$, + and morphisms + \[(f, \phi_{0}, \phi_{1}, A h = k) : + (A, a_{0}, a_{1}, h : a_{0} \to a_{1}) \to + (B, b_{0}, b_{1}, k : b_{0} \to b_{1})\] + where $f : A \to B$, + $\phi_{i} : f a_{i} \to b_{i}$ and $A h = k$ represents a merely propositional + proof of equality. + Then we can also compute + \[\rho' (A, a) = (A, a, a, \id_{a})\] +\end{lemma} + +\begin{lemma} + \label{q_classifies} + + Specialized to $q : I \to \Type$ in $\Pshgrpd$, + the characterizing property of polynomial endofunctors + \cref{polynomial_endofunctor_property} says that a map + from a representable + $\ep : \Ga \to \Poly{q}X$ + corresponds to the data of + \[ + A : \Ga \to \Type + \quad \quad + \text{ and } + \quad \quad + C : \Ga \cdot A \cdot A \cdot \Id \to X + \] + where $A = q \circ \ep$ and +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + X & {\Ga \cdot A \cdot A \cdot \Id} & {I'} & \ptgrpd \\ + & {\Ga \cdot A \cdot A} & {U \times_\grpd U} & \grpd \\ + & {\Ga \cdot A} & \ptgrpd \\ + & \Ga & \grpd + \arrow["C"', from=1-2, to=1-1] + \arrow[from=1-2, to=1-3] + \arrow[from=1-2, to=2-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-2, to=2-3] + \arrow[from=1-3, to=1-4] + \arrow[from=1-3, to=2-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-3, to=2-4] + \arrow["U", from=1-4, to=2-4] + \arrow[from=2-2, to=2-3] + \arrow[from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["{\Id'}"', from=2-3, to=2-4] + \arrow["\fst"', from=2-3, to=3-3] + \arrow[from=3-2, to=3-3] + \arrow[from=3-2, to=4-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-2, to=4-3] + \arrow["U", from=3-3, to=4-3] + \arrow["A"', from=4-2, to=4-3] +\end{tikzcd}\] +\end{lemma} + +\begin{lemma} + \label{T_classifies} + The data of a map $(A, C, \ga_{\refl}) : \Ga \to T$ + corresponds to the data of + \begin{align*} + A \, & : \Ga \to \grpd \\ + C \, & : \Ga \cdot A \cdot A \cdot \Id \to \grpd \\ + \ga_{\refl} \, & : \Ga \cdot A \to \ptgrpd \\ + \text{ such that } \, & C \circ A^{*} \rho' = U \circ \ga_{\refl} \\ + \end{align*} +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + \ptgrpd && {\Ga \cdot A} & \ptgrpd & \ptgrpd \\ + \grpd && {\Ga \cdot A \cdot A \cdot \Id} & {I'} & \grpd \\ + && {\Ga \cdot A \cdot A} & {U \times_\grpd U} & \ptgrpd \\ + && {\Ga \cdot A} & \ptgrpd & \grpd \\ + && \Ga & \grpd + \arrow["U"', from=1-1, to=2-1] + \arrow["{\ga_\refl}"{description}, from=1-3, to=1-1] + \arrow["{\var_A}", from=1-3, to=1-4] + \arrow["{A^* \rho'}"{description}, from=1-3, to=2-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-3, to=2-4] + \arrow["{\rho'}"{description}, from=1-4, to=2-4] + \arrow["U", from=1-5, to=2-5] + \arrow["C"{description}, from=2-3, to=2-1] + \arrow[from=2-3, to=2-4] + \arrow["{\disp{\Id' \circ U^* \var_A}}"', from=2-3, to=3-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-3, to=3-4] + \arrow[from=2-4, to=1-5] + \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=45}, draw=none, from=2-4, to=2-5] + \arrow[from=2-4, to=3-4] + \arrow[from=3-3, to=3-4] + \arrow["{\disp{U \circ \var_A}}"', from=3-3, to=4-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-3, to=4-4] + \arrow["{\Id'}"{description}, from=3-4, to=2-5] + \arrow["\snd"', from=3-4, to=3-5] + \arrow["\fst"{description}, from=3-4, to=4-4] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-4, to=4-5] + \arrow["U"', from=3-5, to=4-5] + \arrow["{\var_A}"{description}, from=4-3, to=4-4] + \arrow["{\disp{A}}"', from=4-3, to=5-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=4-3, to=5-4] + \arrow["U"', from=4-4, to=4-5] + \arrow["U", from=4-4, to=5-4] + \arrow["A"', from=5-3, to=5-4] +\end{tikzcd}\] + Then precomposition with $\si : \De \to \Ga$ + acts on such a triple via +% https://q.uiver.app/#q=WzAsNCxbMiwxLCJUIl0sWzAsMV0sWzEsMSwiXFxHYSJdLFsxLDAsIlxcRGUiXSxbMywyLCJcXHNpIiwyXSxbMywwLCIoQSBcXGNpcmMgXFxzaSwgQyBcXGNpcmMgcV4qIFxcc2ksIFxcZ2FfXFxyZWZsIFxcY2lyYyBcXHRwXipcXHNpICkiXSxbMiwwLCIoQSwgQywgXFxnYV9cXHJlZmwpIiwyXV0= +\[\begin{tikzcd} + & \De \\ + {} & \Ga & T + \arrow["\si"', from=1-2, to=2-2] + \arrow["{(A \circ \si, C \circ q^* \si, \ga_\refl \circ \tp^*\si )}", from=1-2, to=2-3] + \arrow["{(A, C, \ga_\refl)}"', from=2-2, to=2-3] +\end{tikzcd}\] + +\end{lemma} +\begin{proof} +% https://q.uiver.app/#q=WzAsNSxbMCwwLCJcXEdhIl0sWzEsMiwiXFxQb2x5e3F9IHtcXFR5cGV9Il0sWzIsMiwiXFxQb2x5e1xcdHB9IHtcXFRlcm19Il0sWzIsMSwiXFxQb2x5e1xcdHB9IHtcXFRlcm19Il0sWzEsMSwiVCJdLFswLDEsIihBLCBDKSIsMix7ImN1cnZlIjoyLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMSwyLCJcXFN0YXJ7XFxyaG99X3tcXFR5cGV9IiwyXSxbMywyLCJcXFBvbHl7XFx0cH0ge1xcdHB9Il0sWzAsMywiKEEsIFxcZ2FfXFxyZWZsKSIsMCx7ImN1cnZlIjotMywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzQsMV0sWzQsM10sWzQsMiwiIiwyLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV0sWzAsNCwiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d +\[\begin{tikzcd} + \Ga \\ + & T & {\Poly{\tp} {\Term}} \\ + & {\Poly{q} {\Type}} & {\Poly{\tp} {\Term}} + \arrow[dashed, from=1-1, to=2-2] + \arrow["{(A, \ga_\refl)}", bend left, dashed, from=1-1, to=2-3] + \arrow["{(A, C)}"', bend right, dashed, from=1-1, to=3-2] + \arrow[from=2-2, to=2-3] + \arrow[from=2-2, to=3-2] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3] + \arrow["{\Poly{\tp} {\tp}}", from=2-3, to=3-3] + \arrow["{\Star{\rho}_{\Type}}"', from=3-2, to=3-3] +\end{tikzcd}\] + By the universal property of pullbacks, + The data of a map from a respresentable $\Ga \to T$ + corresponds to the data of + $(A, C) : \Ga \to \Poly{q}{\Type}$ and $(A',\ga_\refl) : \Ga \to \Poly{\tp}{\Term}$ + such that + \[ \Star{\rho}_{\Type} \circ (A, C) = \Poly{\tp} {\tp} \circ (A', \ga_\refl) \] + By \cref{polynomial_on_slice_map} and \cref{polynomial_endofunctor_property} + this says + \[ (A, C \circ A^{*} \rho) = (A', \tp \circ \ga_\refl)\] + so the above is equivalent to having $A = A', C, \ga_\refl$ such that + \[ C \circ A^{*} \rho = \tp \circ \ga_\refl \text{ in } \Pshgrpd \] + By Yoneda this is equivalent to requiring + \[ C \circ A^{*} \rho' = U \circ \ga_\refl \text{ in } \Cat \] +\end{proof} + +\begin{prop} \label{ep_computation} + We can compute $\ep : \Poly{q}{\Term} \to T$ via + \begin{align*} + \ep_{\Ga} : \,& \Pshgrpd (\Ga, \Poly{q}{\Term}) \to \Pshgrpd (\Ga, T) \\ + & (A, \ga) \mapsto (A, U \circ \ga, \ga \circ A^{*} \rho') + \end{align*} +\end{prop} +\begin{proof} + This follows from the computation for $T$ \cref{T_classifies}, + the polynomial action on slice morphisms \cref{polynomial_on_slice_map}, and \cref{polynomial_endofunctor_property}. +\end{proof} + +\begin{defn}[Identity elimination] + We want to define $J : T \to \Poly{q}{\Term}$ + \begin{align*} + J_{\Ga} : \Pshgrpd (\Ga, T) \to \, & \Pshgrpd (\Ga, \Poly{q}{\Term})\\ + (A, C, \ga_{\refl}) \mapsto \, & (A, \ga) + \end{align*} + for some + $\ga : \Ga \cdot A \cdot A \cdot \Id \to \ptgrpd$ which we will define below. + We first use $T$ \cref{T_classifies} to describe the given data: +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + \ptgrpd && {\Ga \cdot A} & \ptgrpd & \ptgrpd \\ + \grpd && {\Ga \cdot A \cdot A \cdot \Id} & {I'} & \grpd \\ + && {\Ga \cdot A \cdot A} & {U \times_\grpd U} & \ptgrpd \\ + && {\Ga \cdot A} & \ptgrpd & \grpd \\ + && \Ga & \grpd + \arrow["U"', from=1-1, to=2-1] + \arrow["{\ga_\refl}"{description}, from=1-3, to=1-1] + \arrow["{\var_A}", from=1-3, to=1-4] + \arrow["{A^* \rho'}"{description}, from=1-3, to=2-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-3, to=2-4] + \arrow["{\rho'}"{description}, from=1-4, to=2-4] + \arrow["U", from=1-5, to=2-5] + \arrow["\ga"{description}, dashed, from=2-3, to=1-1] + \arrow["C"{description}, from=2-3, to=2-1] + \arrow[from=2-3, to=2-4] + \arrow["{\disp{\Id' \circ U^* \var_A}}"', from=2-3, to=3-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-3, to=3-4] + \arrow[from=2-4, to=1-5] + \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=45}, draw=none, from=2-4, to=2-5] + \arrow[from=2-4, to=3-4] + \arrow[from=3-3, to=3-4] + \arrow["{\disp{U \circ \var_A}}"', from=3-3, to=4-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-3, to=4-4] + \arrow["{\Id'}"{description}, from=3-4, to=2-5] + \arrow["\snd"', from=3-4, to=3-5] + \arrow["\fst"{description}, from=3-4, to=4-4] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-4, to=4-5] + \arrow["U"', from=3-5, to=4-5] + \arrow["{\var_A}"{description}, from=4-3, to=4-4] + \arrow["{\disp{A}}"', from=4-3, to=5-3] + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=4-3, to=5-4] + \arrow["U"', from=4-4, to=4-5] + \arrow["U", from=4-4, to=5-4] + \arrow["A"', from=5-3, to=5-4] +\end{tikzcd}\] + Let us name the fibers over the diagonal + \[C_{\refl} := U \circ \ga_{\refl} = C \circ A^{*} \rho' : \Ga \cdot A \to \grpd\] + and its given points + \[\ga_{\refl} = (C_{\refl}, c_{\refl})\] + (Note that $c_{\refl}$ is not a functor, but will give us + an object per object $(x, a)$, and morphism + $c_{\refl}(f, \phi) : C_{\refl}(f,\phi) c_{\refl}(x, a) \to c_{\refl} (y, b)$ + per morphism $(f, \phi)$.) + Then $\ga$ will be defined by + using $C$ to lift the path + \[(\id_{x},\id_{a_{0}},h, \_) : (x, a_{0}, a_{0}, \id_{a}) \to (x, a_{0}, a_{1}, h) \in \Ga \cdot A \cdot A \cdot \Id\] + that starts on the diagonal, + to give us a point in any fiber, using $c_{\refl}$. + Note that we unfolded $\Ga \cdot A \cdot A \cdot \Id$ as the domain of the nested + display maps + so that $x \in \Ga$, $a_{0} \in A x$, + \[ a_{1} \in U \circ \var_{A} (x, a_{0}) = U (A x, a_{0}) = A x\] + and + \[ h \in Id' \circ U^{*}\var_{A} (x, a_{0}, a_{1}) = + \Id' (A x, a_{0}, a_{1}) = A x (a_{0}, a_{1})\] + We also check $(\id_{x},\id_{a_{0}},h, \_)$ + is a path in $\Ga \cdot A \cdot A \cdot \Id$ + by proving ``$\_$'', the ommited equality + \[ (\Id' \circ U^{*}\var_{A} (\id_{x}, \id_{a_{0}}, h)) \id_{a_{0}} + = (\Id' (A \id_{x}, \id_{a_{0}}, h)) \id_{a_{0}} + = h \circ A \id_{x} \id_{a_{0}} \circ \id_{a_{0}}^{-1} + = h \] + So we define $\ga : \Ga \cdot A \cdot A \cdot \Id \to \ptgrpd$ on objects by + \[ + (x, a_{0}, a_{1}, h) \mapsto (C (x, a_{0}, a_{1}, h), + C(\id_{x}, \id_{a_{0}}, h, \_) \, c_{\refl}(x, a_{0}))\] + noting that from the computation of $\rho'$ given in + \cref{I'_classifies} it follows that + \[ c_{\refl} (x, a_{0}) \in C \circ A^{*} \rho' (x, a_{0}) + = C(x, a_{0}, a_{1}, h)\] + Define $\ga$ on morphism + $(f, \phi_{0}, \phi_{1}, \phi_{1} \circ A f h \circ \phi_{0}^{-1} = k) : (x, a_{0}, a_{1}, h) \to (y, b_{0}, b_{1}, k)$ + by + \[ (f, \phi_{0}, \phi_{1}, \_) \mapsto + ( + C (f, \phi_{0}, \phi_{1}, \_), + C(\id_{y}, \id_{b_{0}}, k, \_) \, c_{\refl}(f, \phi_{0}) + )\] + We type check + $C(\id_{y}, \id_{b_{0}}, k, \_) \, c_{\refl}(f, \phi_{0})$ + \begin{align*} + C(\id_{y}, \id_{b_{0}}, k, \_) c_{\refl}(f, \phi_{0}) \quad : \quad + & C (f, \phi_{0}, \phi_{1}, \_) \circ + C(\id_{x}, \id_{a_{0}}, h, \_) \, c_{\refl}(x, a_{0}) \\ + = \, & C (f, \phi_{0}, \phi_{1} \circ A f h, \_) \, c_{\refl}(x, a_{0}) \\ + = \, & C (f, \phi_{0}, k \circ \phi_{0}, \_) \, c_{\refl}(x, a_{0}) \\ + = \, & C (\id_{y}, \id_{b_{0}}, k , \_) \circ + C (f, \phi_{0}, \phi_{0}, \_) \, c_{\refl}(x, a_{0}) \\ + = \, & C (\id_{y}, \id_{b_{0}}, k , \_) \circ + C_{\refl} (f, \phi_{0}) \, c_{\refl}(x, a_{0}) \\ + \to \, & C (\id_{y}, \id_{b_{0}}, k , \_) \, c_{\refl} (y, b_{0})\\ + \end{align*} +\end{defn} +\begin{proof} + Functoriality of $\ga$ is routine. + We show naturality of $J$. Suppose $\si : \De \to \Ga$ is representable +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + & {(A \circ \si, C \circ q^* \si, \ga_\refl \circ \tp^*\si )} & {\quad \quad(A \circ \si, \ga_\De) } \\ + {} && {(A \circ \si, \ga_\Ga \circ q^*\si)} \\ + & {\Pshgrpd(\De, T)} & {\Pshgrpd(\De, \Poly{q}{\Term})} \\ + & {\Pshgrpd(\Ga, T)} & {\Pshgrpd(\Ga, \Poly{q}{\Term})} \\ + & {(A, C, \ga_\refl)} & {(A, \ga_\Ga)} + \arrow[maps to, from=1-2, to=1-3] + \arrow[Rightarrow, dashed, no head, from=1-3, to=2-3] + \arrow["{J_\De}", from=3-2, to=3-3] + \arrow["{- \circ \si}", from=4-2, to=3-2] + \arrow["{J_\Ga}"', from=4-2, to=4-3] + \arrow["{- \circ \si}"', from=4-3, to=3-3] + \arrow[maps to, from=5-2, to=5-3] +\end{tikzcd}\] + So we want to show that on objects $(x, a_{0}, a_{1}, h) \in \De \cdot A \circ \si \cdot A \circ \si \cdot \Id$ + \[ \ga_{\De} \, (x, a_{0}, a_{1}, h) = + \ga_{\Ga} \circ q^{*} \si \, (x, a_{0}, a_{1}, h) + \] + Let us denote $q^{*} \si (x, a_{0}, a_{1}, h) = (\si x, a_{0}', a_{1}', h')$. + Then + \begin{align*} + & \ga_{\De} \, (x, a_{0}, a_{1}, h) \\ + = \, & (C \circ q^{*} \si \, (x, a_{0}, a_{1}, h), + (C \circ q^{*} \si \, (\id_{x}, \id_{a_{0}}, h, \_)) + (c_{\refl} (\tp^{*}\si (x, a_{0}))) + )\\ + = \, & (C \, (\si x, a_{0}', a_{1}', h') , + (C \, (\id_{\si x}, \id_{a_{0}'}, h', \_)) + (c_{\refl} \, (\si x, a_{0}')) + )\\ + % = \, & (C \, (\si x, a_{0}', a_{1}', h'),) + = \, & \ga_{\Ga}(\si x, a_{0}', a_{1}', h') \\ + = \, & \ga_{\Ga} \circ q^{*} \si \, (x, a_{0}, a_{1}, h) \\ + \end{align*} + and similarly for morphisms. +\end{proof} + +\begin{prop} + $J : T \to \Poly{q}{\Term}$, as defined above is a section of $\ep$. +\end{prop} +\begin{proof} + Let $(A, C, \ga_{\refl}) : \Ga \to T$ be a map from a representable. + Then using the definition of $J$ and the computation of $\ep$ + \cref{ep_computation} + \[\ep_{\Ga} \circ J_{\Ga} \, (A, C, \ga_{\refl}) + = \ep_{\Ga} (A, \ga) + = (A, U \circ \ga, \ga \circ A^{*} \rho') + \] + By definition of $\ga$ from $J$ we can see that $U \circ \ga = C$, + so it suffices to show that $\ga \circ A^{*} \rho' = \ga_{\refl}$. + On an object $(x, a_{0})$ + \begin{align*} + \ga \circ A^{*} \rho' (x, a_{0}) \\ + = \, & \ga (x, a_{0}, a_{0}, \id_{a_{0}})\\ + = \, & (C(x,a_{0},a_{0},\id_{a_{0}}), C(\id_{x}, \id_{a_{0}}, \id_{a_{0}}) \, c_{\refl})\\ + = \, & (C_{\refl} (x, a_{0}), c_{\refl}(x, a_{0}))\\ + \end{align*} +\end{proof} + diff --git a/blueprint/src/macros/common.tex b/blueprint/src/macros/common.tex index e0396b5..a181c0a 100644 --- a/blueprint/src/macros/common.tex +++ b/blueprint/src/macros/common.tex @@ -86,6 +86,7 @@ \newcommand{\Id}{\mathsf{Id}} \newcommand{\refl}{\mathsf{refl}} \newcommand{\fst}{\mathsf{fst}} +\newcommand{\snd}{\mathsf{snd}} \newcommand{\ev}[2]{\mathsf{ev}_{#1} \, #2} \newcommand{\Fib}{\mathsf{Fib}}