ScalingStacks

Example 5.10. Let $\mathcal{C}$ be an $\mathcal{E}_{1}$-algebra in $\mathcal{P}r^{\rm L}$ (so $\mathcal{C}$ is a presentable monoidal $\infty$-category whose monoidal structure distributes over colimits). Then left modules for the monoidal $\infty$-category $U_{\mathcal{C}}$ are equivalent to $\mathcal{C}$-bimodules. In particular, we have an equivalence $U_{\mathcal{C}}\simeq\mathcal{C}\otimes\mathcal{C}^{\rm op}$, where here $\mathcal{C}^{\rm op}$ denotes the $\infty$-category $\mathcal{C}$ equipped with the opposite monoidal structure. As a consequence, we see that in this case, the preceding general definition of $\mathcal{F}$-Hochschild cohomology recaptures the notion of the Drinfeld center introduced earlier.