ScalingStacks

Definition 1.4. Let $A$ be an associative algebra object in a closed symmetric monoidal $\infty$-category $\mathcal{S}$.

1. (1)

   The derived center or Hochschild cohomology $\mathcal{Z}(A)=\hh^{*}(A)\in\mathcal{S}$ is the endomorphism object ${\mathcal{E}nd}_{A\otimes A^{\rm op}}(A)$ of $A$ as an $A$-bimodule.

2. (2)

   The derived trace or Hochschild homology $\mathcal{T}r(A)=\hh_{*}(A)\in\mathcal{S}$ is the pairing object ${A\otimes_{A\otimes A^{\rm op}}A}$ of $A$ with itself as an $A$-bimodule.