# Kerodon

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Corollary 4.5.7.2. Let $\operatorname{\mathcal{W}}$ denote the full subcategory of $\operatorname{Fun}( , \operatorname{Set_{\Delta }})$ spanned by those morphisms of simplicial sets $f: X \rightarrow Y$ which are categorical equivalences. Then $\operatorname{\mathcal{W}}$ is closed under the formation of filtered colimits in $\operatorname{Fun}( , \operatorname{Set_{\Delta }})$.

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 $X$, the simplicial set $Q(X)$ is an $\infty$-category and the morphism $u_ X: X \rightarrow Q(X)$ is inner anodyne. For every morphism of simplicial sets $f: X \rightarrow Y$, we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [r]^-{f} \ar [d]^{ u_ X } & Y \ar [d]^{ u_{Y} } \\ Q(X) \ar [r]^-{ Q(f) } & Q(Y) }$

where the vertical maps are categorical equivalences (Corollary 4.5.3.14). It follows from Remark 4.5.3.5 that $f$ is a categorical equivalence if and only if the functor $Q(f)$ is an equivalence of $\infty$-categories. Using the criterion of Theorem 4.5.7.1, we see that $f$ is a categorical equivalence if and only if the induced map $\operatorname{Fun}( \Delta ^1, Q(X) )^{\simeq } \rightarrow \operatorname{Fun}( \Delta ^1, Q(Y) )^{\simeq }$ is a homotopy equivalence of Kan complexes. The desired result now follows by observing that the construction $X \mapsto \operatorname{Fun}( \Delta ^1, Q(X) )^{\simeq }$ commutes with filtered colimits, since the collection of homotopy equivalences between Kan complexes is closed under filtered colimits (Proposition 3.2.8.3). $\square$