Example 5.3.4.24 (Homotopy Quotients). Let $G$ be a group and let $BG$ denote the associated groupoid (consisting of a single object with automorphism group $G$). Let $X$ be a simplicial set equipped with an action of $G$, which we identify with a functor $\mathscr {F}: BG \rightarrow \operatorname{Set_{\Delta }}$. Applying Corollary 5.3.4.22, we obtain an isomorphism of simplicial sets $X_{\mathrm{h}G} \xrightarrow {\sim } \operatorname{N}_{\bullet }^{\mathscr {F}}(BG)$, where $X_{\mathrm{h}G}$ is the homotopy quotient of $X$ by the action of $G$ (Example 5.3.2.15). If $X$ is a Kan complex, then Corollary 5.3.4.23 guarantees that $X_{\mathrm{h}G}$ is also a Kan complex.

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