Kerodon

Remark 9.1.7.2. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $W$ denote the collection of horizontal edges of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$. Proposition 9.1.7.1 asserts that, if the category $\operatorname{\mathcal{C}}$ is filtered, then the comparison map $\theta : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} )$ exhibits $\varinjlim ( \mathscr {F} )$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ with respect to $W$. In particular, $\theta $ is a weak homotopy equivalence.