Kerodon

Example 7.5.7.2. Let $\operatorname{\mathcal{C}}$ be a groupoid and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ is a Kan complex (Corollary 5.3.4.23). In this case, Proposition 7.5.7.1 guarantees that $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is a colimit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ in the $\infty $-category $\operatorname{\mathcal{S}}$. For example, if $X$ is a Kan complex equipped with an action of a group $G$, then the homotopy quotient $X_{\mathrm{h}G}$ is a colimit of the associated diagram $B_{\bullet }G \rightarrow \operatorname{\mathcal{S}}$ (Example 5.3.4.24).