Kerodon

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

Definition 7.1.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that an object $Y \in \operatorname{\mathcal{C}}$ is initial if, for every object $Z \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is a contractible Kan complex. We say that $Y$ is final if, for every object $X \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is a contractible Kan complex.