Kerodon

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

Example 4.6.6.7. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, so that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.4.5.1). Combining Remark 4.6.6.6 with Theorem 4.6.7.5, we deduce the following:

  • An object $Y \in \operatorname{\mathcal{C}}$ is initial when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Z \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$ is contractible.

  • An object $Y \in \operatorname{\mathcal{C}}$ final when viewed as an object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is contractible.