ScalingStacks

Unlike in classical algebra, for an algebra object in an $\infty$-category, commutativity is an additional structure rather than a property. More precisely, the forgetful functor from $\mathcal{E}_{\infty}$-algebras to $\mathcal{E}_{1}$-algebras is conservative but not fully faithful. Given an $\mathcal{E}_{\infty}$-algebra $A$ in a symmetric monoidal $\infty$-category $\mathcal{S}$, the center $\mathcal{Z}(A)$ that we have studied to this point does not involve the commutativity of $A$. More precisely, it only depends upon the $\mathcal{E}_{1}$-algebra underlying $A$.

In this section, we discuss higher versions of the center where we forget less commutativity. For a perfect stack $X$, we calculate the $\mathcal{E}_{n}$-center of the $\mathcal{E}_{n}$-category underlying the $\mathcal{E}_{\infty}$-category $\qc(X)$. A detailed treatment of the foundations of the subject may be found in Chapter 2 of [F1]. Here we summarize only what is required for our consideration of the $\mathcal{E}_{n}$-center of $\qc(X)$.

Let $F$ be a topological operad. We define a topological category $\mathcal{F}$ with objects finite pointed sets and morphism spaces given by

We then obtain an $\infty$-category, also denoted by $\mathcal{F}$, by applying the singular functor to the mapping spaces to get a simplicial category and then taking the simplicial nerve.

We will refer to the $\infty$-category $\mathcal{C}$ equipped with an $\mathcal{F}$-monoidal structure as an $\mathcal{F}$-category. This construction will be of particular interest to us when $\mathcal{F}$ is the little $n$-disks operad $\mathcal{E}_{n}$.

To discuss Hochschild cohomology, we must next introduce the notion of operadic modules.

To this end, let ${{\rm Fin}_{*}}$ be the category of finite based sets. Let ${\rm Fin}_{+}$ be the category of “doubly-based sets” with objects finite based sets $I_{*}$ and $(I\amalg+)_{*}$, and morphisms $\alpha$ of the underlying based sets such that $+\in\alpha^{-1}(+)$, and either $\alpha(+)=+$ or $\alpha(+)=*$.

Given a topological operad $F$, recall the $\infty$-category $\mathcal{F}$ introduced above. We also define the $\infty$-category $\mathcal{F}_{+}$ to be the fiber product of $\infty$-categories

There is a similar definition of an $\mathcal{F}$-algebra and an $\mathcal{F}$-$A$-module in a symmetric monoidal or $\mathcal{F}$-monoidal $\infty$-category $\mathcal{M}$ generalizing the definition given above. This requires the notion of an $\mathcal{F}$-lax monoidal functor (see [F1, Chapter 2, Definition 3.10]). The simplification of the previous definition is available because ${\rm Cat}_{\infty}$ is equipped with the Cartesian monoidal structure. We will avail ourselves of this greater generality in the following definition, although the only case that will concern us in the following is when $\mathcal{M}$ is either ${\rm Cat}_{\infty}$ or $\mathcal{P}r^{\rm L}$.

Now let $\mathcal{M}$ be a presentable symmetric monoidal $\infty$-category whose monoidal structure distributes over colimits. Let $A$ be an $\mathcal{F}$-algebra in $\mathcal{M}$, and let $\mathrm{Mod}_{A}^{\mathcal{F}}(\mathcal{M})$ be the $\infty$-category of $\mathcal{F}$-$A$-modules in $\mathcal{M}$ (see [F1, Chapter 2, Definition 4.3]). Under the above assumptions, the $\infty$-category $\mathrm{Mod}_{A}^{\mathcal{F}}(\mathcal{M})$ is naturally tensored over $\mathcal{M}$.

In particular, when $\mathcal{M}=\mathcal{P}r^{\rm L}$, the $\mathcal{F}$-Hochschild cohomology of an $\mathcal{F}$-category $\mathcal{C}$ is the $\infty$-category of $\mathcal{F}$-$\mathcal{C}$-module functors

We now specialize to the case where $\mathcal{F}$ is the $\mathcal{E}_{n}$-operad. Intuitively, an $\mathcal{E}_{n}$-algebra structure on an object $A$ is equivalent to a family of associative algebra structures on $A$ parameterized by $S^{n-1}$ such that antipodal points are associated to opposite multiplications on $A$. An $\mathcal{E}_{n}$-$A$-module structure on $M$ admits a similar intuitive interpretation as a family of compatible left $A$-module structures on $M$ parameterized by $S^{n-1}$. This intuition leads to the following (to appear in [F2], see also [L7]).

For $n=1$, the proposition reduces to the familiar statement that for an $\mathcal{E}_{\infty}$-algebra $A$, an $A\otimes A^{\rm op}$-module structure (in the form of an $\mathcal{E}_{1}$-$A$-module structure) is equivalent to an $A\otimes A$-module structure (in the form of a left $U_{A}$-module structure).

For our current purposes, one can interpret the proposition as furnishing the definition of an $\mathcal{E}_{n}$-$A$-module. Namely, the reader uncomfortable with the abstractions can take left $S^{n-1}\otimes A$-modules as the definition of $\mathcal{E}_{n}$-$A$-modules

Let $A$ be an $\mathcal{E}_{\infty}$-algebra and $U_{A}$ be as above. We have a companion definition of $\mathcal{E}_{n}$-Hochschild homology.

We now have the following contribution of this paper to the story.