Kerodon

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

Example 5.1.6.2. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, so that the projection maps $U: \operatorname{\mathcal{C}}\rightarrow \Delta ^0$ and $V: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ are inner fibrations. Then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories if and only if it is an equivalence of inner fibrations over $\Delta ^0$. In particular, the inner fibrations $U$ and $V$ are equivalent (in the sense of Definition 5.1.6.1) if and only if the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are equivalent (in the sense of Definition 4.5.1.10).