Kerodon

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

Definition 1.3.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that $f$ is an isomorphism if the homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. We will say that two objects $X,Y \in \operatorname{\mathcal{C}}$ are isomorphic if there exists an isomorphism from $X$ to $Y$ (that is, if $X$ and $Y$ are isomorphic as objects of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$).