8.3.7 Hom-Functors for Oriented Fiber Products
Let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, which we regard as fixed throughout this section. Let $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ be the oriented fiber product of Definition 4.6.4.1, so that we have projection maps $\operatorname{ev}_{-}: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{-}$ and $\operatorname{ev}_{+}: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{+}$. Recall that objects of $\operatorname{\mathcal{C}}_{\pm }$ can be identified with triples $C = (C_{-}, C_{+}, \alpha _{C})$, where $C_{-} = \operatorname{ev}_{-}(C)$ is an object of $\operatorname{\mathcal{C}}_{-}$, $C_{+} = \operatorname{ev}_{+}(C)$ is an object of $\operatorname{\mathcal{C}}_{+}$, and $\alpha _{C}: F_{-}(C_{-}) \rightarrow F_{+}(C_{+})$ is a morphism in the $\infty $-category $\operatorname{\mathcal{C}}$. Let $D = (D_{-}, D_{+}, \alpha _{D})$ be another object of $\operatorname{\mathcal{C}}_{\pm }$. Our goal in this section is to show that there is a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{\pm }}( C, D ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}_+}( C_+, D_+ ) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{-}}( C_{-}, D_{-} ) \ar [r] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( F_{-}( C_{-}), F_+(D_+) ) } \]
in the $\infty $-category $\operatorname{\mathcal{S}}$. Moreover, this pullback diagram can be chosen to depend functorially on both $C$ and $D$.
Theorem 8.3.7.1. Let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(\bullet , \bullet ): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{\mathcal{C}}$, and define $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{-} }(\bullet , \bullet )$, $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_+}(\bullet , \bullet )$, and $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{\pm } }(\bullet , \bullet )$ similarly. Then there is a pullback diagram
8.57
\begin{equation} \begin{gathered}\label{equation:Hom-in-oriented-fiber-product} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{\pm }}( \bullet , \bullet ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{+} }( \operatorname{ev}_{+}( \bullet ), \operatorname{ev}_{+}( \bullet ) ) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{-}}( \operatorname{ev}_{-}(\bullet ), \operatorname{ev}_{-}(\bullet ) ) \ar [r] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( (F_{-} \circ \operatorname{ev}_-)(\bullet ), (F_{+} \circ \operatorname{ev}_+)(\bullet ) ) } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{\pm }, \operatorname{\mathcal{S}})$.
Our proof of Theorem 8.3.7.1 will require several auxiliary constructions.
Notation 8.3.7.3. By definition, the oriented fiber product $\operatorname{\mathcal{C}}_{\pm } = \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ is universal among $\infty $-categories equipped with functors $\operatorname{ev}_{-}: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{-}$ and $\operatorname{ev}_{+}: \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{+}$, together with a natural transformation $\alpha : F_{-} \circ \operatorname{ev}_{-} \rightarrow F_{+} \circ \operatorname{ev}_{+}$ of functors from $\operatorname{\mathcal{C}}_{\pm }$ to $\operatorname{\mathcal{C}}$. Let us identify $\alpha $ with a functor $\Delta ^1 \times \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}$. Applying the twisted arrow construction, we obtain a functor $\operatorname{Tw}(\alpha ): \operatorname{Tw}(\Delta ^1) \times \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$, which we can identify with a diagram $\Xi : \operatorname{Tw}(\Delta ^1) \rightarrow \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}_{\pm } ), \operatorname{Tw}(\operatorname{\mathcal{C}}) )$.
Recall that objects of $\operatorname{Tw}(\Delta ^1)$ can be identified with ordered pairs $(i,j)$ satisfying $0 \leq i \leq j \leq 1$. By construction, $\Xi (0,0)$ is the functor $\operatorname{Tw}( F_{-} \circ \operatorname{ev}_{-} ): \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$, and $\Xi (1,1)$ is the functor $\operatorname{Tw}( F_{+} \circ \operatorname{ev}_{+} ): \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$. Setting $T = \Xi (0,1)$, we obtain a pair of natural transformations $\operatorname{Tw}( F_{-} \circ \operatorname{ev}_{-} ) \xrightarrow {\xi _{-}} T \xleftarrow { \xi _{+} } \operatorname{Tw}( F_{+} \circ \operatorname{ev}_{+} )$. The triple $( \operatorname{Tw}( \operatorname{ev}_{-} ), T, \xi _{-} )$ determines a functor $Q_{-}: \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \rightarrow \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}})$, and the triple $( \operatorname{Tw}( \operatorname{ev}_{+} ), T, \xi _{+} )$ determines a functor $Q_{+}: \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}})$.
The diagonal embedding $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ and its opposite determine monomorphisms of simplicial sets
\[ \delta _{-}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \hookrightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} \quad \quad \delta _{+}: \operatorname{\mathcal{C}}_{+} \hookrightarrow \operatorname{\mathcal{C}}_{+} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\quad \quad C_{+} \mapsto ( C_{+}, F_{+}(C_{+}), \operatorname{id}). \]
We let $\operatorname{\mathcal{E}}_{-} \subseteq \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}})$ denote the inverse image of the image of $\delta _{-}$ and $\operatorname{\mathcal{E}}_{+} \subseteq \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) )$ the inverse image of the image of $\delta _{+}$, so that we have pullback diagrams of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{-} \ar [r] \ar [d] & \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d] & \operatorname{\mathcal{E}}_{+} \ar [r] \ar [d] & \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{C}}_{-} \ar [r]^{ \delta _{-} } & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} & \operatorname{\mathcal{C}}_{+} \ar [r]^{ \delta _{+} } & \operatorname{\mathcal{C}}_{+} \vec{\times }_{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}. } \]
Note that the functors $Q_{-}$ and $Q_{+}$ factor through $\operatorname{\mathcal{E}}_{-}$ and $\operatorname{\mathcal{E}}_{+}$, respectively. Moreover, projection onto the second factor determines morphisms $\pi _{-}: \operatorname{\mathcal{E}}_{-} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$ and $\pi _{+}: \operatorname{\mathcal{E}}_{+} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$ satisfying $\pi _{-} \circ Q_{-} = T = \pi _{+} \circ Q_{+}$, so that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [r]^{ Q_{+} } \ar [d]^{Q_{-}} & \operatorname{\mathcal{E}}_{+} \ar [d]^{ \pi _{+} } \\ \operatorname{\mathcal{E}}_{-} \ar [r]^{ \pi _{-} } & \operatorname{Tw}(\operatorname{\mathcal{C}}). } \]
Our first goal is to analyze the simplicial sets $\operatorname{\mathcal{E}}_{-}$ and $\operatorname{\mathcal{E}}_{+}$ appearing in Notation 8.3.7.3. The twisted arrow fibration for $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ determine morphisms
\[ \overline{U}_{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \vec{\times }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow ( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} ) \times ( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \]
\[ \overline{U}_{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}_{+}) \vec{\times }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow ( \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} ) \times ( \operatorname{\mathcal{C}}_{+} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}). \]
Restricting to the inverse images of $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times ( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})$ and $( \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} ) \times \operatorname{\mathcal{C}}_{+}$, respectively, we obtain morphisms
\[ U_{-}: \operatorname{\mathcal{E}}_{-} \rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times ( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \quad \quad U_{+}: \operatorname{\mathcal{E}}_{+} \rightarrow ( \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} ) \times \operatorname{\mathcal{C}}_{+}. \]
Lemma 8.3.7.4. The morphism $U_{-}: \operatorname{\mathcal{E}}_{-} \rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times ( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}})$ is a left fibration, with covariant transport representation given by the composition
\[ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times ( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{-} \xrightarrow { \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{-} }( \bullet , \bullet ) } \operatorname{\mathcal{S}}. \]
Similarly, $U_{+}$ is a left fibration with covariant transport representation given by the composition
\[ ( \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} ) \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{C}}_{+}}( \bullet , \bullet )} \operatorname{\mathcal{S}}. \]
Proof.
Projection onto the first factor determines morphism $\operatorname{\mathcal{E}}_{-} \subseteq \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}_{-})$ which fits into a commutative diagram
8.58
\begin{equation} \begin{gathered}\label{equation:E-plusminus-covariant-transport} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{-} \ar [d]^{ U_{-} } \ar [r] & \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ) \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times ( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \ar [r] & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{-}. } \end{gathered} \end{equation}
To prove the first assertion, it will suffice to show that the diagram (8.58) induces a trivial Kan fibration
\[ \theta _{-}: \operatorname{\mathcal{E}}_{-} \rightarrow ( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times ( \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) ) \times _{ ( \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{-} )} \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ). \]
Unwinding the definitions, we see that $\theta _{-}$ is a pullback of the map
\[ \overline{\theta }_{-}: \operatorname{Fun}( \Delta ^1, \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}) } \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}) \]
induced by the projection map $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$. Since $\lambda $ is a left fibration (Proposition 8.1.1.11), the morphism $\overline{\theta }_{-}$ is a trivial Kan fibration (Proposition 4.2.6.1), so that $\theta _{-}$ is also a trivial Kan fibration. The proof of the second assertion is similar.
$\square$
The main content of Theorem 8.3.7.1 is contained in the following:
Proposition 8.3.7.6. The diagram of $\infty $-categories
8.59
\begin{equation} \begin{gathered}\label{equation:Hom-in-oriented-fiber-product-reformulated} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [r]^{ Q_{+} } \ar [d]^{Q_{-}} & \operatorname{\mathcal{E}}_{+} \ar [d]^{ \pi _{+} } \\ \operatorname{\mathcal{E}}_{-} \ar [r]^{ \pi _{-} } & \operatorname{Tw}(\operatorname{\mathcal{C}}) } \end{gathered} \end{equation}
is a categorical pullback square (see Notation 8.3.7.3).
Proof.
Using Corollary 4.5.2.24, we can factor the functor $F_{-}$ as a composition $\operatorname{\mathcal{C}}_{-} \xrightarrow { G } \operatorname{\mathcal{C}}'_{-} \xrightarrow { F'_{-} } \operatorname{\mathcal{C}}$, where $G$ is an equivalence of $\infty $-categories and $F'_{-}$ is an isofibration. Set $\operatorname{\mathcal{C}}'_{\pm } = \operatorname{\mathcal{C}}'_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ and define $\operatorname{\mathcal{E}}'_{-} \subseteq \operatorname{Tw}( \operatorname{\mathcal{C}}'_{-} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}})$ as in Notation 8.3.7.3. Then (8.59) can be identified with the outer rectangle in a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [d]^{Q_{-}} \ar [r] & \operatorname{Tw}( \operatorname{\mathcal{C}}'_{\pm } ) \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{+} \ar [d]^{\pi _{+}} \\ \operatorname{\mathcal{E}}_{-} \ar [r] & \operatorname{\mathcal{E}}'_{-} \ar [r] & \operatorname{Tw}(\operatorname{\mathcal{C}}). } \]
It follows from Remark 4.6.4.4 and Corollary 8.1.2.16 that the horizontal map on the upper left is an equivalence of $\infty $-categories. The proof of Lemma 8.3.7.4 shows that the horizontal map on the lower left is also an equivalence of $\infty $-categories, so the left half of the diagram is a categorical pullback square (Proposition 4.5.2.22). Consequently, to show that the outer rectangle is a categorical pullback square, it will suffice to show that the right half of the diagram is a categorical pullback square (Proposition 4.5.2.19). We may therefore replace $\operatorname{\mathcal{C}}_{-}$ by $\operatorname{\mathcal{C}}'_{-}$, and thereby reduce to proving Proposition 8.3.7.6 in the special case where $F_{-}$ is an isofibration of $\infty $-categories. Similarly, we may assume that $F_{+}$ is an isofibration of $\infty $-categories. In this case, the functors $\pi _{-}$ and $\pi _{+}$ are also isofibrations. We are therefore reduced to proving that the functors $Q_{-}$ and $Q_{+}$ induce an equivalence of $\infty $-categories $Q: \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \rightarrow \operatorname{\mathcal{E}}_{-} \times _{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}_{+}$ (see Proposition 4.5.2.27).
The functor $Q$ fits into a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [r]^{Q} \ar [d] & \operatorname{\mathcal{E}}_{-} \times _{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}_{+} \ar [r] \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ) \times \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [d]^{ \operatorname{Tw}(F_{-} ) \times \operatorname{Tw}(F_{+}) } \\ \operatorname{Tw}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \ar [r]^{\overline{Q}} & \operatorname{Fun}_{\pm }( \operatorname{Tw}(\Delta ^1), \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \ar [r] & \operatorname{Tw}(\operatorname{\mathcal{C}}) \times \operatorname{Tw}(\operatorname{\mathcal{C}}), } \]
where $E$ is the equivalence of $\infty $-categories appearing in Example 8.2.4.8. Since the outer rectangle and right half of this diagram are pullback square, the left half of the diagram is also a pullback square. Consequently, to show that $Q$ is an equivalence of $\infty $-categories, it will suffice to show that the vertical maps are isofibrations of $\infty $-categories (Corollary 4.5.2.33). This follows from Corollary 8.1.1.16, by virtue of our assumption that $F_{-}$ and $F_{+}$ are isofibrations.
$\square$
To deduce Theorem 8.3.7.1 from Proposition 8.3.7.6, we will use the following general observation:
Lemma 8.3.7.7. Suppose we are given a commutative diagram of $\infty $-categories
8.60
\begin{equation} \begin{gathered}\label{equation:covariant-transport-categorical-pullback-square} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{00} \ar [dr]^{ U_{00} } \ar [rr] \ar [dd] & & \operatorname{\mathcal{E}}_{01} \ar [dd] \ar [dr]^{U_{01}} & \\ & \operatorname{\mathcal{D}}_{00} \ar [rr] \ar [dd] & & \operatorname{\mathcal{D}}_{01} \ar [dd] \\ \operatorname{\mathcal{E}}_{10} \ar [dr]^{U_{10} } \ar [rr] & & \operatorname{\mathcal{E}}_{11} \ar [dr]^{U_{11} } & \\ & \operatorname{\mathcal{D}}_{10} \ar [rr] & & \operatorname{\mathcal{D}}_{11} } \end{gathered} \end{equation}
satisfying the following conditions:
- $(1)$
The front face $\sigma $:
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}_{00} \ar [r] \ar [d] & \operatorname{\mathcal{D}}_{01} \ar [d] \\ \operatorname{\mathcal{D}}_{10} \ar [r] & \operatorname{\mathcal{D}}_{11} } \]
is a categorical pullback square.
- $(2)$
The back face $\tau :$
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{00} \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{01} \ar [d] \\ \operatorname{\mathcal{E}}_{10} \ar [r] & \operatorname{\mathcal{E}}_{11} } \]
is a categorical pullback square.
- $(3)$
Each of the functors $U_{ij}: \operatorname{\mathcal{E}}_{ij} \rightarrow \operatorname{\mathcal{D}}_{ij}$ is a left fibration with covariant transport representation $\mathscr {F}_{ij}: \operatorname{\mathcal{D}}_{ij} \rightarrow \operatorname{\mathcal{S}}$.
Then there exists a pullback square
\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}_{00} \ar [r] \ar [d] & \mathscr {F}_{01} |_{ \operatorname{\mathcal{D}}_{00} } \ar [d] \\ \mathscr {F}_{10} |_{ \operatorname{\mathcal{D}}_{00} } \ar [r] & \mathscr {F}_{11} |_{ \operatorname{\mathcal{D}}_{00} } } \]
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{D}}_{00}, \operatorname{\mathcal{S}})$.
Proof.
Let us regard $\sigma $ and $\tau $ as functors from the partially ordered set $[1] \times [1]$ to the category of simplicial sets, and let $\operatorname{\mathcal{D}}= \operatorname{N}_{\bullet }^{\sigma }( [1] \times [1] )$ and $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\tau }( [1] \times [1] )$ denote the weighted nerves of Definition 5.3.3.1. The diagram (8.60) can be identified with a natural transformation from $\tau $ to $\sigma $, which induces a functor of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$. By construction, we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^{ U } \ar [dr] & & \operatorname{\mathcal{D}}\ar [dl] \\ & \Delta ^1 \times \Delta ^1, & } \]
where the vertical maps are cocartesian fibrations (Corollary 5.3.3.16). In what follows, we will identify each $\operatorname{\mathcal{D}}_{ij}$ and $\operatorname{\mathcal{E}}_{ij}$ with the full subcategories of $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ given by the fiber of the vertex $(i,j) \in \Delta ^1 \times \Delta ^1$, so that $U_{ij} = U|_{ \operatorname{\mathcal{D}}_{ij} }$.
It follows from Corollary 5.3.3.18 that $U$ is a left fibration. Let $\mathscr {F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $U$. Without loss of generality, we may assume that each $\mathscr {F}_{ij}$ coincides with the restriction $\mathscr {F}|_{ \operatorname{\mathcal{D}}_{ij} }$. Let $\widetilde{\sigma }: [1] \times [1] \rightarrow \operatorname{Set_{\Delta }}$ be the constant functor taking the value $\operatorname{\mathcal{D}}_{00}$. There is a unique natural transformation $\widetilde{\sigma } \rightarrow \sigma $ which is the identity when evaluated at $(0,0) \in [1] \times [1]$, which induces a functor
\[ T: \operatorname{\mathcal{D}}_{00} \times \Delta ^1 \times \Delta ^1 \simeq \operatorname{N}_{\bullet }^{\widetilde{\sigma }}( [1] \times [1] ) \rightarrow \operatorname{N}_{\bullet }^{\sigma }( [1] \times [1] ) = \operatorname{\mathcal{D}}. \]
We can then identify the composition $\mathscr {F} \circ T$ with a commutative diagram
8.61
\begin{equation} \begin{gathered}\label{equation:covariant-transport-categorical-pullback-square2} \xymatrix@R =50pt@C=50pt{ \mathscr {F}_{00} \ar [r] \ar [d] & \mathscr {F}_{01} |_{ \operatorname{\mathcal{D}}_{00} } \ar [d] \\ \mathscr {F}_{10} |_{ \operatorname{\mathcal{D}}_{00} } \ar [r] & \mathscr {F}_{11} |_{ \operatorname{\mathcal{D}}_{00} } } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{D}}_{00}, \operatorname{\mathcal{S}})$. We will complete the proof by showing that this diagram is a (levelwise) pullback square.
Fix an object $D_{00} \in \operatorname{\mathcal{D}}_{00}$, having image $D_{ij} \in \operatorname{\mathcal{D}}_{ij}$ for each pair $i,j \in [1]$. Evaluation at $D_{00}$ then carries (8.61) to a diagram
8.62
\begin{equation} \begin{gathered}\label{equation:covariant-transport-categorical-pullback-square3} \xymatrix@R =50pt@C=50pt{ \mathscr {F}_{00}(D_{00} ) \ar [r] \ar [d] & \mathscr {F}_{01}( D_{01} ) \ar [d] \\ \mathscr {F}_{10}( D_{10} ) \ar [r] & \mathscr {F}_{11}( D_{11} ) } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{S}}$, and we wish to show that (8.62) is a pullback square. Using Example 5.6.5.6, we can identify (8.62) with the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \tau _0 )$, where $\tau _0: [1] \times [1] \rightarrow \operatorname{Kan}$ is the commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ U_{00}^{-1}( D_{00} ) \ar [r] \ar [d] & U_{01}^{-1}( D_{01} ) \ar [d] \\ U_{10}^{-1}( D_{10} ) \ar [r] & U_{11}^{-1}( D_{11} ). } \]
By construction, we have a pullback diagram
8.63
\begin{equation} \begin{gathered}\label{equation:covariant-transport-categorical-pullback-square4} \xymatrix@R =50pt@C=50pt{ \tau _0 \ar [r] \ar [d] & \tau \ar [d] \\ \underline{\Delta ^0} \ar [r] & \sigma } \end{gathered} \end{equation}
in the category $\operatorname{Fun}( [1] \times [1], \operatorname{\mathcal{QC}})$. For each pair $i,j \in [1]$, assumption $(3)$ guarantees that $U_{ij}: \operatorname{\mathcal{E}}_{ij} \rightarrow \operatorname{\mathcal{D}}_{ij}$ is an isofibration of $\infty $-categories, so that evaluation at $(i,j)$ carries (8.63) to a categorical pullback square of $\infty $-categories. Applying Example 7.6.3.4, we obtain a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \tau _0 ) \ar [r] \ar [d] & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \tau ) \ar [d] \\ \underline{ \Delta ^0 } \ar [r] & \operatorname{N}_{\bullet }^{\operatorname{hc}}( \sigma ) } \]
in the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{QC}})$. Assumptions $(2)$ and $(3)$ guarantee that $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\sigma )$ and $\operatorname{N}_{\bullet }(\tau )$ are limit diagrams in the $\infty $-category $\operatorname{\mathcal{QC}}$ (Example 7.6.3.4). Applying Corollary 7.3.8.7, we conclude that $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \tau _0 )$ is also a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$, and therefore also in the full subcategory $\operatorname{\mathcal{S}}\subset \operatorname{\mathcal{QC}}$.
$\square$
Proof of Theorem 8.3.7.1.
Let $U_{\pm }: \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \rightarrow \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{\pm }$ and $U: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ be the projection maps of Notation 8.1.1.6. We have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [rr]^{ Q_{+} } \ar [dr]^{ U_{\pm } } \ar [dd]^{Q_{-}} & & \operatorname{\mathcal{E}}_{+} \ar [dr]^{ U_{+} } \ar [dd] & \\ & \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{\pm } \ar [rr] \ar [dd] & & ( \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} ) \times \operatorname{\mathcal{C}}_{+} \ar [dd] \\ \operatorname{\mathcal{E}}_{-} \ar [rr] \ar [dr]^{ U_{-} } & & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [dr]^{U} & \\ & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times (\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \ar [rr] & & \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, } \]
where the diagonal maps are left fibrations whose covariant transport representations are given by (the restrictions of) $\operatorname{Hom}$-functors for the $\infty $-categories $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}_{\pm }$ (Lemma 8.3.7.4). By virtue of Lemma 8.3.7.7, it will suffice to show that the front and back faces of this diagram are categorical pullback squares. For the back face, this follows from Proposition 8.3.7.6. The front face is a cartesian product of a pair of pullback diagrams
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \ar [r] \ar [d] & \operatorname{\mathcal{C}}^{\operatorname{op}}_{+} \vec{\times }_{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{\mathcal{C}}^{\operatorname{op}} \ar [d] & \operatorname{\mathcal{C}}_{\pm } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{+} \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \ar [r] & \operatorname{\mathcal{C}}^{\operatorname{op}} & \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}. } \]
We are therefore reduced to showing that these pullback squares are categorical pullback squares (Remark 4.5.2.10). This follows from Corollary 4.5.2.28, since vertical maps of the left square and horizontal maps of the right square are isofibrations (Proposition 4.6.4.2).
$\square$