Definition 4.8.7.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. We say that $F$ is categorically $n$-connective if it is $m$-full for every nonnegative integer $m \leq n$ (see Definition 4.8.5.10).
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