Proposition 4.8.8.10. Let $n$ be an integer and let $\operatorname{\mathcal{W}}_ n$ denote the full subcategory of $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$ spanned by those morphisms of simplicial sets $f: A \rightarrow B$ which are categorically $n$-connective. Then $\operatorname{\mathcal{W}}_ n$ is closed under the formation of filtered colimits in $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Corollary 4.1.3.3, there exists a functor $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ which commutes with filtered colimits and a natural transformation of functors $u: \operatorname{id}_{\operatorname{Set_{\Delta }}} \rightarrow Q$ with the property that, for every simplicial set $A$, the simplicial set $Q(A)$ is an $\infty $-category and the morphism $u_ A: A \rightarrow Q(A)$ is inner anodyne. For every morphism of simplicial sets $f: A \rightarrow B$, we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{ u_{A} } & Q(A) \ar [d]^{ Q(f) } \\ B \ar [r]^-{ u_{B} } & Q(B) } \]
where the horizontal maps are categorical equivalences. It follows that $f$ is categorically $n$-connective if and only if $Q(f)$ is categorically $n$-connective. The desired result now follows from the analogous statement for filtered colimits of $\infty $-categories (Remark 4.8.5.11). $\square$