# Kerodon

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Corollary 5.3.4.23. Let $\operatorname{\mathcal{C}}$ be a groupoid and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Then the homotopy colimit $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is a Kan complex.

Proof. Using Corollaries 5.3.4.22 and 5.3.3.16, we see that the map $U: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration. Since $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a Kan complex (Proposition 1.2.4.2), it follows that $U$ is a Kan fibration (Corollary 4.4.3.8), so that $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ is also a Kan complex (Remark 3.1.1.11). $\square$