Symmetric monoidal structure on module categories

Theorem 19.2.14 ([Lurie (2017), Theorem 4.5.2.1]). Let \(C\) be a symmetric monoidal \(\infty \)-category. Assume that \(C\) admits geometric realizations and that the tensor product of \(C\) preserves geometric realizations in both variables. Let \(R\) be a commutative algebra in \(C\). Then:

(1)

The \(\infty \)-category \(\Mod _R(C)\) admits a symmetric monoidal structure.

(2)

The functor \(\Mod _R(C) \to {}_R\BMod _R(C)\) from Observation 19.2.4 refines to a monoidal functor, where the target has the monoidal structure from Proposition 19.2.13.

In particular, the tensor product on \(\Mod _R(C)\) is given as the following composite: \[ \begin {aligned} \Mod _R(C) \times \Mod _R(C) &\longrightarrow {}_R\BMod _R(C) \times {}_R\BMod _R(C) \\ &\xrightarrow {- \otimes _R -} {}_R\BMod _R(C) \longrightarrow \LMod _R(C) \iso \Mod _R(C). \end {aligned} \]