Kerodon

Remark 4.8.5.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. It follows from Remark 4.8.5.5 (and Theorem 4.8.4.1) that if $F$ is categorically $(n+1)$-connective, then the induced map of homotopy $n$-categories $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}})}} \rightarrow \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{D}})}}$ is an equivalence. In particular, if $F$ is categorically $2$-connective, then it induces an equivalence of homotopy categories $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$.