Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.5.7.7. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Then $\overline{\mathscr {F}}$ is a homotopy colimit diagram (in the sense of Definition 7.5.7.3) if and only if the induced functor of $\infty $-categories

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \overline{\mathscr {F}}): \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\triangleright } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}} \]

is a colimit diagram (in the sense of Variant 7.1.2.5).

Proof. Combine Proposition 7.5.7.6 with Corollary 7.5.4.6 (applied to the simplicial category $\operatorname{Kan}^{\operatorname{op}}$). $\square$