Notation 8.3.7.3 (06C2)

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{-} \ar [r] \ar [d] & \operatorname{Tw}( \operatorname{\mathcal{C}}_{-} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d] & \operatorname{\mathcal{E}}_{+} \ar [r] \ar [d] & \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \ar [r]^-{ \delta _{-} } & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} & \operatorname{\mathcal{C}}_{+} \ar [r]^-{ \delta _{+} } & \operatorname{\mathcal{C}}_{+} \vec{\times }_{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}. } \]

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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [r]^-{ Q_{+} } \ar [d]^{Q_{-}} & \operatorname{\mathcal{E}}_{+} \ar [d]^{ \pi _{+} } \\ \operatorname{\mathcal{E}}_{-} \ar [r]^-{ \pi _{-} } & \operatorname{Tw}(\operatorname{\mathcal{C}}). } \]