Kerodon

Warning 4.7.3.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$ which is final as an object of the homotopy category $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$. Then $X$ need not be final 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 final when viewed as an object of the homotopy category $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} = \pi _{\leq 1}(\operatorname{\mathcal{C}})$. However, $X$ is final as an object of $\operatorname{\mathcal{C}}$ only if $\operatorname{\mathcal{C}}$ is contractible (Corollary 4.7.3.13).