Remark 4.8.9.4 (05DL)

Remark 4.8.9.4. Let $n$ be an integer. The notion of categorical $n$-connectivity is completely determined by the following two properties:

$(1)$

If $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{B}}$ is a functor of $\infty $-categories, then it is categorically $n$-connective in the sense of Definition 4.8.9.2 if and only if it is categorically $n$-connective in the sense of Definition 4.8.7.1: that is, $F$ is $m$-full for every nonnegative integer $m \leq n$ (see Proposition 4.8.9.1).

$(2)$

Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { A \ar [r] \ar [d]^{f} & A' \ar [d]^{f'} \\ B \ar [r] & B', } \]

where the horizontal maps are categorical equivalences. Then $f$ is categorically $n$-connective if and only if $f'$ is categorically $n$-connective. See Proposition 4.5.2.19.

If $f: A \rightarrow B$ is any morphism of simplicial sets, then we can use Proposition 4.1.3.2 to choose a commutative diagram

\[ \xymatrix { A \ar [d]^{f} \ar [r] & \operatorname{\mathcal{A}}\ar [d]^{F} \\ B \ar [r] & \operatorname{\mathcal{B}}} \]

where the horizontal maps are categorical equivalences and $F$ is a functor of $\infty $-categories. Combining $(1)$ and $(2)$, we see that $f$ is categorically $n$-connective if and only if the functor $F$ is $m$-full for $m \leq n$.