Corollary 9.2.9.5. Let $n$ be an integer, let $S_{L}$ denote the collection of all $n$-connective morphisms between Kan complexes, and let $S_{R}$ denote the collection of all $(n-1)$-truncated morphisms between Kan complexes. Then the pair $(S_ L, S_ R)$ determines a factorization system on the $\infty $-category $\operatorname{\mathcal{S}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Recall that a morphism of Kan complexes is $n$-connective if and only if it is categorically $n$-connective (Example 4.8.7.3), and $(n-1)$-truncated if and only if it is essentially $(n-1)$-categorical (Example 4.8.6.3). It follows immediately from Proposition 9.2.9.4 that $S_{L}$ and $S_{R}$ are closed under isomorphism, and that $S_{L}$ is left orthogonal to $S_{R}$. To complete the proof, it suffices to show that every morphism of Kan complexes $f: X \rightarrow Z$ admits a factorization $X \xrightarrow {f_{L}} Y \xrightarrow {f_{R}} Z$, where $f_{L}$ is $n$-connective and $f_{R}$ is $(n-1)$-truncated. This is the content of Corollary 4.8.8.9. $\square$