Corollary 24.2.3. For every finite category \(I\), the functor \(\Hom _{\Cat _\infty }(I,-)\colon \Cat _{\infty } \to \An \) preserves filtered colimits. In particular the core functor \((-)^{\simeq }\colon \Cat _{\infty } \to \An \) preserves filtered colimits.
Proof. By definition, the subcategory \(\Cat ^{\fin }_{\infty } \subseteq \Cat _{\infty }\) is the smallest subcategory closed under pushouts which contains \(\emptyset \), \([0]\) and \([1]\). Since filtered colimits commute with pullbacks in \(\An \), the class \(I\) of \(\infty \)-categories for which the claim holds is closed under pushouts. It thus remains to show the claim for \(I = \emptyset , [0],[1]\). The claim for \(\emptyset \) is clear, since \(\Hom _{\Cat _{\infty }}(\emptyset ,C)\) is terminal for each \(C\) and filtered colimits of terminal objects are terminal. The claim for \([0]\) and \([1]\) follows from the previous lemma, since \(\Hom _{\Cat _{\infty }}([n],-)\) agrees with the composite \(\Cat _{\infty } \xrightarrow {N} \sAn \xrightarrow {\ev _n} \An \).
The last claim is the special case \(I = [0]\). □
Generated from the authoritative LaTeX source.