Lemma 4.2 (0NK8) · ScalingStacks

0NK8

Lemma 4.2. $\Psi_{\tri}{\mathcal{C}}$ factors through the subcategory $\Cat_{\infty}^{\ex}\subset\Cat_{\infty}$ , and $\Psi_{\perf}{\mathcal{C}}$ factors through the subcategory $\Cat_{\infty}^{\perf}\subset\Cat_{\infty}$ .

0NK9

Proof. As noted in remark 2.13, since $\psi_{\tri}$ is the underlying simplicial category associated to a pretriangulated spectral category, $\Psi_{\perf}$ is characterized as the idempotent-completion of $\psi_{\tri}$ given by proposition 3.2 coupled with corollary 3.3. Finally, note that maps of spectral categories induce, by left Kan extension, finite colimit-preserving on the level of stable $\infty$ -categories. ∎