Theorem 4.23 (0NL8) · ScalingStacks

Theorem 4.23. The functor

\Psi_{\perf}\colon\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\infty}^{\perf}

admits a fully faithful and accessible right adjoint

\Upsilon\colon\Cat_{\infty}^{\perf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}].

That is, the $\infty$ -category of idempotent-complete stable $\infty$ -categories is an accessible localization of the $\infty$ -category of spectral categories obtained by inverting the Morita equivalences.

0NL9

Proof. The $\infty$ -category of idempotent-complete stable $\infty$ -categories is a localizing subcategory of the $\infty$ -category of stable $\infty$ -categories, and idempotent-completion is an accessible functor as the inclusion $\Psi_{\tri}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Psi_{\perf}$ preserves filtered colimits. ∎