ScalingStacks

Remark 4.2.9.

As in observation 4.2.4, any presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}\)-enriched \(\infty\)-category is additive and any presentably \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\)-enriched \(\infty\)-category is stable, and we have the following equivalences: \[\begin{aligned} \mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}& \coloneqq \mathrm{Mod}_{\mathrm{Mod}^{\geq 0,\mathcal Z}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L}) \simeq \mathrm{Mod}_{\mathrm{Mod}^{\geq 0,\mathcal Z}_{\mathbb{K}}}({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}}) \\ {\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}}}& \coloneqq \mathrm{Mod}_{\mathrm{Mod}^{\mathcal Z}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L})\simeq\mathrm{Mod}_{\mathrm{Mod}^{\mathcal Z}_{\mathbb{K}}}({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}) \end{aligned}\] Similarly, we have the following equivalences for their small variants, abbreviating \(M=\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}\): \[\begin{aligned} \mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{{\mathrm{Mod}_{\mathbb{K}}^{\geq 0,\mathcal Z}}}& \coloneqq\mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0, \mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) \simeq \mathrm{Mod}_{M}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}) \simeq \mathrm{Mod}_{M^{\mathrm{cp}}}(\mathrm{add}) \simeq \mathrm{Mod}_{M^{\mathrm{cp}}}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}) \\ \mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}&\coloneqq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}) \simeq \mathrm{Mod}_{\left(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\right)^{\mathrm{c}}}(\mathrm{st})\simeq \mathrm{Mod}_{\left(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\right)^{\mathrm{c}}}(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}) \end{aligned}\]