Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 8.5.1.27. Let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in \operatorname{\mathcal{I}}}$ be a diagram of simplicial sets indexed by a filtered category $\operatorname{\mathcal{I}}$. Suppose that each $\operatorname{\mathcal{C}}_{i}$ is an $\infty $-category. Then the tautological map

\[ \theta : \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{\mathcal{C}}_ i ) \]

is an equivalence of $\infty $-categories.

Proof. The morphism $\theta $ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \operatorname{\mathcal{C}}_ i ) \ar [r]^-{\theta } \ar [d] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Ret}), \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{\mathcal{C}}_ i ) \ar [d] \\ \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Fun}( \mathcal{R}, \operatorname{\mathcal{C}}_ i ) \ar [r]^-{\theta '} & \operatorname{Fun}( \mathcal{R}, \varinjlim _{i \in \operatorname{\mathcal{I}}} \operatorname{\mathcal{C}}_ i ), } \]

where the vertical maps are trivial Kan fibrations (Corollary 8.5.1.26). It will therefore suffice to show that $\theta '$ is an equivalence of $\infty $-categories. In fact, $\theta '$ is an isomorphism of simplicial sets, since the simplicial set $\mathcal{R}$ is finite (Corollary 3.5.1.10). $\square$