Kerodon

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

Definition 4.5.2.1. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. We say that $f$ is a categorical equivalence if, for every $\infty $-category $\operatorname{\mathcal{C}}$, the induced functor $\operatorname{Fun}( Y, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})$ induces a bijection on isomorphism classes $\pi _0( \operatorname{Fun}(Y,\operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } )$.