Kerodon

Remark 7.4.4.4. Let $\operatorname{\mathcal{C}}$ be a small simplicial set, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram, and let $\varprojlim (\mathscr {F} )$ denote its limit (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 $\varprojlim (\mathscr {F} )$ 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 limits. This is a special case of Variant 7.1.4.25, since $\operatorname{\mathcal{S}}$ is a reflective subcategory of $\operatorname{\mathcal{QC}}$ (Example 6.2.2.6). However, it can also be deduced directly from Corollary 7.4.4.2: 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), so that $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) = \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \int _{\operatorname{\mathcal{C}}} \mathscr {F} )$ is a Kan complex by virtue of Corollary 4.4.2.5.