ScalingStacks

Observation 4.2.6.

As \(\infty\)-categories of modules of commutative algebras in presentably symmetric monoidal \(\infty\)-categories, both \(\mathrm{add}_{\mathbb{K}}\) and \(\mathrm{st}_{\mathbb{K}}\) are presentably symmetric monoidal. The symmetric monoidal structure on \(\mathrm{add}_{\mathbb{K}}\) can be characterized as follows: for \(\mathcal C, \mathcal D\in \mathrm{add}_{\mathbb{K}}\), there is a functor \(\mathcal C\times \mathcal D\rightarrow\mathcal C\otimes \mathcal D\) which is additive and \(\mathbb{K}\)-linear in either variable, and which for all \(\mathcal E\in \mathrm{add}_{\mathbb{K}}\) induces an equivalence between the \(\infty\)-category of additive \(\mathbb{K}\)-linear functors \(\mathcal C\otimes \mathcal D\rightarrow\mathcal E\) and the \(\infty\)-category of functors \(\mathcal C\times \mathcal D\rightarrow\mathcal E\) that are additive and \(\mathbb{K}\)-linear in either variable. An analogous characterization with additive replaced by exact holds for \(\mathrm{st}_{\mathbb{K}}\).