ScalingStacks

Observation 4.2.4.

Since \(\mathrm{Mod}_{\mathbb{K}}\) is stable and \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0}\) is additive, we obtain the following equivalences from proposition 3.1.8.([0034]): \[\begin{aligned} {\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{add}_{\mathbb{K}}}}}&:= \mathrm{Mod}_{\mathrm{Mod}^{\geq 0}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L}) \simeq \mathrm{Mod}_{\mathrm{Mod}^{\geq 0}_{\mathbb{K}}}({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{add}}})\\ {\mathrm{Pr}^{\mathrm{L}}_{{\mathrm{st}_{\mathbb{K}}}}}& := \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^\mathrm{L})\simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}({\mathrm{Pr}^{\mathrm{L}}_{\mathrm{st}}}) \end{aligned}\] In particular, any presentably \(\mathrm{Mod}_{\mathbb{K}}\)-enriched \(\infty\)-category is automatically stable, and any presentably \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0}\)-enriched \(\infty\)-category is automatically additive. Combining proposition 3.1.8.([0034]) with the equivalences from §3.1 and subsection 3.3, we obtain the analogous characterizations of their small variants: \[\begin{aligned} \mathrm{add}_{\mathbb{K}}& :=\mathrm{Mod}_{\mathrm{CProj}_{\mathbb{K}}}(\mathrm{add}) \simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0}}(\mathrm{Pr}^{\mathrm{L}, \mathrm{cp}}_{\mathrm{add}}) \simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}^{\geq 0}}(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}) \simeq \mathrm{Mod}_{\mathrm{CProj}_{\mathbb{K}}}(\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}) \\ \mathrm{st}_{\mathbb{K}}&:=\mathrm{Mod}_{\mathrm{Perf}_{\mathbb{K}}}(\mathrm{st})\simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^{\mathrm{L}, \mathrm{c}}_{\mathrm{st}}) \simeq \mathrm{Mod}_{\mathrm{Mod}_{\mathbb{K}}}(\mathrm{Pr}^{\mathrm{L},\mathrm{c}}) \simeq \mathrm{Mod}_{\mathrm{Perf}_{\mathbb{K}}}(\mathrm{Cat}_{\infty}^{\mathrm{rex}, \mathrm{idem}}) \end{aligned}\]