ScalingStacks

We will apply the above constructions in the following setting. We will always take $\mathcal{S}$ to be the $\infty$-category ${\mathcal{P}r}^{\rm L}$ of presentable $\infty$-categories (with morphisms left adjoints). Then a monoidal presentable $\infty$-category $\mathcal{C}$ is an associative algebra object in $\mathcal{P}r^{\rm L}$. Thus we have the notion of its center (or Hochschild cohomology category) $\mathcal{Z}(\mathcal{C})=\Fun_{\mathcal{C}\otimes\mathcal{C}^{\rm op}}(\mathcal{C},\mathcal{C})\in\mathcal{P}r^{\rm L}$, and its trace (or Hochschild homology category) $\mathcal{T}r(\mathcal{C})=\mathcal{C}\otimes_{\mathcal{C}\otimes\mathcal{C}^{\rm op}}\mathcal{C}\in\mathcal{P}r^{\rm L}$.

Now we will specialize further to a geometric setting. Let $X$ be a perfect stack, and take $\mathcal{C}$ to be the presentable stable $\infty$-category $\qc(X)$ equipped with its (symmetric) monoidal tensor product. To calculate the center and trace of $\qc(X)$, we introduce the loop space

where the fiber product is along two copies of the diagonal map.