Kerodon

Warning 4.5.1.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. If $F$ is an equivalence of $\infty $-categories (in the sense of Definition 4.5.1.10), then it is a homotopy equivalence of simplicial sets (in the sense of Definition 3.1.6.1). More precisely, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is a homotopy inverse to the functor $F$ (in the sense of Definition 4.5.1.10), then $G$ is also a simplicial homotopy inverse to $F$ (in the sense of Definition 3.1.6.1). Beware that the converse assertion is false in general. For example, the projection map $\Delta ^1 \rightarrow \Delta ^0$ is a homotopy equivalence of simplicial sets (with homotopy inverse given by the inclusion $\Delta ^0 \simeq \{ 0\} \hookrightarrow \Delta ^1$), but not an equivalence of $\infty $-categories.