Lemma 5.16 (0NM4) · ScalingStacks

0NM4

Lemma 5.16. Let ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ be an exact functor of small stable $\infty$ -categories. Then the induced map $\Ind({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind({\mathcal{B}})$ of presentable stable $\infty$ -categories preserves $\kappa$ -compact objects for all infinite regular cardinals $\kappa$ .

0NM5

Proof. Recall that a right adjoint preserves $\kappa$ -filtered colimits if and only if its left adjoint preserves $\kappa$ -compact objects [52, 5.5.1.4]. Since functors which preserve filtered colimits also preserve $\kappa$ -filtered colimits and $\Ind(-)$ takes exact functors to functors which preserve compact objects, the result follows. ∎