Proposition 4.8.9.10. Let $n \geq 0$ be an integer and let $f: A \rightarrow B$ be a morphism of simplicial sets which is bijective on simplices of dimension $< n$ and surjective on $n$-simplices. Then $f$ is categorically $n$-connective.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

**Proof.**
Using Proposition 1.1.4.12, we can choose a simplicial subset $A' \subseteq \operatorname{sk}_{n}(A)$ which contains the $(n-1)$-skeleton of $A$, such that $f$ restricts to an isomorphism of $A'$ with the $n$-skeleton of $B$. It follows from Example 4.8.9.9 $f|_{A'}$ is categorically $n$-connective, and that the inclusion map $A' \hookrightarrow A$ is categorically $(n-1)$-connective. Applying Remark 4.8.9.6, we deduce that $f$ is categorically $n$-connective.
$\square$