ScalingStacks

Proposition 4.1. Let $\mathcal{C}$ be a stable presentable symmetric monoidal $\infty$-category, and $A\in\mathcal{C}$ an associative algebra object.

1. (1)

   For any $\mathcal{C}$-module $\mathcal{M}$, there is a canonical equivalence of $\infty$-categories

   $$\Mod_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}\simeq\Mod_{A}(\mathcal{M}).$$

2. (2)

   For $A^{\prime}\in\mathcal{C}$ a second associative algebra, there is a canonical equivalence of $\infty$-categories

   $$\Mod_{A\otimes A^{\prime}}(\mathcal{C})\simeq\Mod_{A}(\mathcal{C})\otimes_{\mathcal{C}}\Mod_{A^{\prime}}(\mathcal{C}).$$

3. (3)

   The $\infty$-category of modules $\Mod_{A}(\mathcal{C})$ is dualizable as a $\mathcal{C}$-module with dual given by the $\infty$-category of modules $\Mod_{A^{\rm op}}(\mathcal{C})$ over the opposite algebra.