Kerodon

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

Example 7.5.4.2 (Limits of Isofibrant Diagrams). Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be a limit diagram in the category of simplicial sets. Suppose that the diagram $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$ is isofibrant (Definition 4.5.6.3) and that, for each object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is a Kan complex. Then $\overline{\mathscr {F}}$ is a homotopy limit diagram of Kan complexes: this follows by combining Corollary 4.5.6.18 with Proposition 7.5.3.12.