ScalingStacks

Remark 4.2.2.

In other words, an additive/stable presentable \(\mathbb{K}\)-linear \(\infty\)-category is an additive/stable presentable \(\infty\)-category \(\mathcal C\) with an action by \(\mathrm{Mod}_{\mathbb{K}}^{\geq 0}\)/\(\mathrm{Mod}_{\mathbb{K}}\), so that the action functor \(\mathrm{Mod}_{\mathbb{K}}^{(\geq 0)} \times \mathcal C\rightarrow\mathcal C\) preserves small colimits in both variables. A small additive/stable idempotent-complete \(\mathbb{K}\)-linear \(\infty\)-category is a small, additive/stable idempotent complete \(\infty\)-category \(\mathcal C\) with an action by \(\mathrm{CProj}_{\mathbb{K}}\) or \(\mathrm{Perf}_\mathbb{K}\), respectively so that the action functor \(\mathrm{CProj}_{\mathbb{K}} \times \mathcal C\rightarrow\mathcal C\) is additive in either variable, or so that the action functor \(\mathrm{Perf}_{\mathbb{K}} \times \mathcal C\rightarrow\mathcal C\) is exact in either variable, respectively.