Kerodon

Remark 7.4.5.4. Let $\operatorname{\mathcal{C}}$ be a small simplicial set, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram, and let $\varinjlim (\mathscr {F} )$ denote its colimit (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$). Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is a Kan complex. Then $\varinjlim (\mathscr {F} )$ is also a Kan complex, which can be regarded as a limit of $\mathscr {F}$ in the subcategory $\operatorname{\mathcal{S}}\subset \operatorname{\mathcal{QC}}$. In particular, the inclusion functor $\operatorname{\mathcal{S}}\hookrightarrow \operatorname{\mathcal{QC}}$ preserves small colimits. This is a special case of Variant 7.1.4.25, since $\operatorname{\mathcal{S}}$ is a coreflective subcategory of $\operatorname{\mathcal{QC}}$ (Example 6.2.2.6). However, it can also be deduced directly from Corollary 7.4.5.2: the assumption that each $\mathscr {F}(C)$ is a Kan complex guarantees that the projection map $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration (Example 5.6.2.9). In particular, every edge of the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is $U$-cocartesian (Example 5.1.1.3), so that the localization $(\int _{\operatorname{\mathcal{C}}} \mathscr {F})[W^{-1}]$ is automatically a Kan complex (Proposition 6.3.1.20).