Example 5.6.2.20 (05Q3): Elements of Diagram $\infty $-Categories

Example 5.6.2.20 (Elements of Diagram $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram. Fix another simplicial set $K$, and let $\mathscr {F}^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ denote the composition of $\mathscr {F}$ with the functor

\[ \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}\quad \quad \operatorname{\mathcal{D}}\mapsto \operatorname{Fun}(K, \operatorname{\mathcal{D}}). \]

Let $\underline{K}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{S}})$ denote the constant map taking the value $K$, so that Remark 5.6.2.18 and Example 5.6.2.17 supply a comparison map $\rho : K \times \operatorname{\mathcal{C}}\rightarrow \int _{\operatorname{\mathcal{C}}} \underline{K}_{\operatorname{\mathcal{C}}}$. The composition

\begin{eqnarray*} K \times \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{K} & \xrightarrow { \rho \times \operatorname{id}} & \int _{\operatorname{\mathcal{C}}} \underline{K}_{\operatorname{\mathcal{C}}} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{K} \\ & \simeq & \int _{\operatorname{\mathcal{C}}} (\underline{K}_{\operatorname{\mathcal{C}}} \times \mathscr {F}^{K}) \\ & \xrightarrow {\operatorname{ev}} & \int _{\operatorname{\mathcal{C}}} \mathscr {F} \end{eqnarray*}

is determines a commutative diagram

\[ \xymatrix { \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{K} \ar [dr]_{U} \ar [rr]^{T} & & \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \ar [dl]^{V} \\ & \operatorname{\mathcal{C}}. & } \]

Here $U$ and $V$ are cocartesian fibrations, and Remark 5.6.2.14 guarantees that $T$ carries $U$-cocartesian edges to $V$-cocartesian edges. Moreover, for each vertex $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $T_{C}: U^{-1} \{ C\} \rightarrow V^{-1} \{ C\} $ fits into a commutative diagram

\[ \xymatrix { \mathscr {F}^{K}(C) \ar@ {=}[r] \ar [dr] & \operatorname{Fun}(K, \mathscr {F}(C) ) \ar [d] \\ \{ C\} \times _{ \operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}^{K} \ar [r]^{T_ C } & \operatorname{Fun}(K, \{ C \} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} ), } \]

where the vertical maps are equivalences by virtue of Example 5.6.2.19. It follows that $T$ is an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$ (see Proposition 5.1.7.15).