Kerodon

Warning 4.8.7.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. If $F$ is categorically $n$-connective, then it is an $n$-connective morphism of simplicial sets (Corollary 4.8.7.17). Beware that the converse is false in general. For example, the projection map $\Delta ^1 \twoheadrightarrow \Delta ^0$ is a homotopy equivalence (and therefore $n$-connective for every integer $n$) which is not categorically $2$-connective.