Proposition 5.31 (0NMR) · ScalingStacks

0NMR

Proposition 5.31. Any map of the form ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Idem({\mathcal{A}})$ is a $\kappa$ -filtered colimit of elements of ${\mathcal{E}}^{\kappa}_{\mathrm{L}}$ .

0NMS

Proof. Write ${\mathcal{A}}\simeq\colim_{\alpha}{\mathcal{A}}_{\alpha}$ as a $\kappa$ -filtered colimit of $\kappa$ -compact small stable $\infty$ -categories ${\mathcal{A}}_{\alpha}$ . Then $\Idem({\mathcal{A}})\simeq\colim_{\alpha}\Idem({\mathcal{A}}_{\alpha})$ , since $\Idem$ (viewed as an endofunctor of $\Cat_{\infty}^{\ex}$ ) commutes with $\kappa$ -filtered colimits — this follows from the characterization of $\Idem$ in terms of a subcategory of the $\Ind$ category [52, 5.4.2.4] and the fact that filtered colimits in $\Cat_{\infty}^{\ex}$ can be computed in $\Cat_{\infty}$ [53, 1.1.4.6]. ∎