Corollary 9.1.3.4. Let $\operatorname{\mathcal{C}}$ be a small filtered $\infty $-category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a $\operatorname{\mathcal{C}}$-indexed diagram of spaces. Then the colimit $\varinjlim (\mathscr {F}) \in \operatorname{\mathcal{S}}$ is contractible if and only if the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is filtered.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$