Lemma 2.20 (0NJU) · ScalingStacks

0NJU

Lemma 2.20. $\Cat_{\infty}^{\perf}$ is a reflective subcategory of $\Cat_{\infty}^{\ex}$ , and the localization functor $\Idem\colon\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}$ is given by the formula $\Idem({\mathcal{C}})\simeq\Ind({\mathcal{C}})^{\omega}$ .

0NJV

Proof. The subcategory of compact objects $\Ind({\mathcal{C}})^{\omega}$ of $\Ind({\mathcal{C}})$ is an idempotent-complete stable $\infty$ -category, so that $\Idem$ is indeed a functor $\Cat_{\infty}^{\ex}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}$ . Now for small stable $\infty$ -categories ${\mathcal{C}}$ and ${\mathcal{D}}$ with ${\mathcal{D}}$ idempotent-complete, we have a commuting square

in which the horizontal maps are the inclusions of the full subcategories of functors which preserve compact objects, and the right vertical map is an equivalence as the natural map $\Ind({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind(\Idem({\mathcal{C}}))$ is an equivalence. Hence $\Ind({\mathcal{C}})^{\omega}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind(\Idem({\mathcal{C}}))^{\omega}$ is an equivalence, and thus the left vertical map is as well. ∎