Kerodon

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

Warning 4.6.6.19. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$ which is initial as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Then $X$ need not be initial when viewed as an object of $\operatorname{\mathcal{C}}$. For example, if $\operatorname{\mathcal{C}}$ is simply connected Kan complex, then every object $X \in \operatorname{\mathcal{C}}$ is initial when viewed as an object of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} = \pi _{\leq 1}(\operatorname{\mathcal{C}})$. However, $X$ is initial as an object of $\operatorname{\mathcal{C}}$ only if $\operatorname{\mathcal{C}}$ is contractible (Corollary 4.6.6.12).