ScalingStacks

A basic operation on associative algebras is the calculation of their center. The derived version of the center of an associative algebra $A$ is the Hochschild cochain complex (or simply the Hochschild cohomology), which calculates the derived endomorphisms of $A$ as an $A$-bimodule. Another basic operation is the calculation of the universal trace (i.e., the universal target for a map out of $A$ coequalizing left and right multiplication). The derived version of the universal trace is the Hochschild chain complex (or the Hochschild homology), which calculates the derived tensor product of $A$ with itself as an $A$-bimodule.

In Section 5.1, we extend the notion of Hochschild homology and cohomology to associative (or $\mathcal{E}_{1}$-)algebra objects in arbitrary closed symmetric monoidal $\infty$-categories. (As with any structure in an $\infty$-category, an associative multiplication, or $\mathcal{E}_{1}$-structure, is a homotopy coherent notion.) In the case of chain complexes, we recover the usual Hochschild chain and cochain complexes. In the case of spectra, we recover topological Hochschild homology and cohomology.

We show in particular that $\hh^{*}(A)$ and $\hh_{*}(A)$ are calculated in this generality by a version of the usual Hochschild complexes, the cyclic bar construction.

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

These are again presentable $\infty$-categories, or in other words, objects of the $\infty$-category $\mathcal{P}r^{\rm L}$. The center $\mathcal{Z}(\mathcal{C})$ comes equipped with a universal central functor to $\mathcal{C}$, and the trace $\mathcal{T}r(\mathcal{C})$ receives a universal trace functor from $\mathcal{C}$.

Now let us return to a geometric setting and consider a perfect stack $X$ and the presentable $\infty$-category $\qc(X)$. Since $\qc(X)$ is symmetric monoidal, it defines a commutative (or $\mathcal{E}_{\infty}$-)algebra object in $\mathcal{P}r^{\rm L}$, and so in particular, an associative algebra object.

To calculate the center and trace of $\qc(X)$, we introduce the loop space

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

For example, when $X$ is an ordinary smooth scheme over a field of characteristic zero, the loop space $\mathcal{L}X$ is the total space $T_{X}[-1]=\Spec\operatorname{Sym}\Omega_{X}[1]$ of the shifted tangent bundle of $X$. When $X$ is the classifying space $BG$ of a group $G$, the loop space $\mathcal{L}BG$ is the adjoint quotient $G/G$.

In Section 5.1, as a corollary of our main technical results, we obtain the following.

Note that the theorem in particular identifies the Hochschild homology and cohomology objects associated to the monoidal $\infty$-category $\qc(X)$. While such an identification may initially appear surprising, it is a natural consequence of the self-duality of $\qc(X)$ over $\qc(X\times X)$ which in turn is a simple consequence of Theorem 1.2.

In Section 5.3, we also discuss a generalization of the theorem to $\mathcal{E}_{n}$-centers and traces when $n>1$. First, we introduce the notion of center and trace for an $\mathcal{E}_{n}$-algebra object in $\mathcal{P}r^{\rm L}$, for any $n$. Since $\qc(X)$ is an $\mathcal{E}_{\infty}$-algebra object in $\mathcal{P}r^{\rm L}$, it is also an $\mathcal{E}_{n}$-algebra object, for any $n$.

We show that the $\mathcal{E}_{n}$-center and trace of $\qc(X)$ are equivalent to the $\infty$-category of quasi-coherent sheaves on the derived mapping space

For example, when $X$ is an ordinary smooth scheme over a field of characteristic zero, the $n$-sphere space $X^{S^{n}}$ is the total space $T_{X}[-n]=\Spec\operatorname{Sym}\Omega_{X}[n]$ of the shifted tangent bundle of $X$. In particular, as we vary $n$, the $\mathcal{E}_{n}$-centers and traces of $\qc(X)$ differ from $\qc(X)$, though they all have $t$-structures with heart the abelian category of quasi-coherent sheaves on $X$.

When $X$ is the classifying space $BG$ of a group $G$, the $n$-sphere space $BG^{S^{n}}$ can be interpreted as the derived stack ${\mathcal{L}oc}_{G}(S^{n})$ of $G$-local systems on $S^{n}$. In particular, when $n=2$, the $\mathcal{E}_{2}$-center and trace of $\qc(BG)$ is the $\infty$-category $\qc({\mathcal{L}oc}_{G}(S^{2}))$ which appears in the Geometric Langlands program (see for example, the work of Bezrukavnikov and Finkelberg [BeF] who identify $\qc({\mathcal{L}oc}_{G}(S^{2}))$ with the derived Satake (or spherical Hecke) category of arc-group equivariant constructible sheaves on the affine Grassmannian for the dual group).