ScalingStacks

A.9.3 Module categories of commutative algebras[00J7]

If \(\mathcal C\) is a symmetric monoidal \(\infty\)-category and \(A\) a commutative algebra in \(\mathcal C\), there is an equivalence \(\mathrm{LMod}_A(\mathcal C) \simeq \mathrm{RMod}_A(\mathcal C)\) treating the given left action as a right action and vice versa. For this reason, we denote the \(\infty\)-category of modules of a commutative algebra simply by \(\mathrm{Mod}_A(\mathcal C)\) and refer to it as the \(\infty\)-category of \(A\)-modules. Moreover, treating an \(A\)-module as a bimodule induces a functor \(\mathrm{Mod}_A(\mathcal C) \rightarrow{}_A\mathrm{BMod}_A(\mathcal C)\). If \(\mathcal C\) is presentably symmetric monoidal, the relativ tensor product \(-\otimes_A -\) defines a presentably monoidal structure on \({}_A\mathrm{BMod}_A(\mathcal C)\). This lifts to a presentably symmetric monoidal structure on \(\mathrm{Mod}_A\) [Lur17, Thm. 4.5.2.1].