Relationship between Hochschild cohomology and Drinfeld centers

Question

Let $HH_*(A,N)$ (or $HH^*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra $A$ with coefficients in an $A$-bimodule $N$.

I was reading nlab's entry on Hochschild cohomology and I saw this term:

Hochschild homology object of any bimodule over an monoid in a monoidal ($\infty ,1$)-category

, which, when $N = A$, is also called the ($\infty ,1$)- or derived center of $A$.

In this paper (http://arxiv.org/pdf/0805.0157v5.pdf) they say that for an associative algebra object $A$ in a closed symmetric monoidal $\infty$-category $\mathcal{S}$, the derived center or Hochschild cohomology $Z(A) = HH^*(A) \in \mathcal{S}$ is the endomorphism object $End_{A \otimes A^{op}} (A)$ of $A$ as an $A$-bimodule.

Then I found this on page 45:

6.2. Deligne-Kontsevich conjectures for derived centers. The notion of Drinfeld center for monoidal stable categories is a categorical analogue of Hochschild cohomology of associative (or $A_{\infty}$ algebras)...

Honestly I had never heard of Hochschild cohomology objects (or derived centers?) or Hochschild (co)homlogy used in this type of context but mostly I wanted to know:

What is the relationship between Hochschild cohomology and Drinfeld centers?

Answer 1

Classically, Hochschild cohomology is an invariant defined for associative algebras while the Drinfeld centre is an invariant defined for monoidal categories. The latter is a categorification of the classical notion of centre of a monoid. In particular the Drinfeld centre of the category with a single object and endomorphism ring $A$ is nothing but $Z(A)$, the centre of $A$ in the usual sense (or more precisely the category with a single object and endomorphism ring $Z(A)$). The terminology originates from the fact that the Drinfeld centre can also be viewed as a categorification of the Drinfeld double construction for Hopf algebras, which goes back to [Vladimir Drinfeld, Quantum groups, Proceedings of the ICM, pp. 798–820, AMS, 1987]. It is easy to see that the definition of Drinfeld centre in fact gives a notion of centre for an associative algebra object in any symmetric monoidal category.

The relationship between these two constructions is that Hochschild homology is a derived version of the centre. That is, the Hochschild homology of an associative algebra $A$ is the centre of $A$ as an associative algebra object in $D(A \otimes A^{op})$, the derived category of $A \otimes A^{op}$-modules. However, to make this precise, one needs to introduce the centre in the setting of $(\infty,1)$-categories, since considering only the homotopy category of the $(\infty,1)$-category $D(A \otimes A^{op})$ loses too much information.

This is the subject of the paper of Ben-Zvi, Francis and Nadler: the generalization of centres to $(\infty,1)$-category theory. Thus they consider the notion of derived centre of an associative algebra object in any symmetric monoidal $(\infty,1)$-category. By specializing this gives a notion of derived centre of an associative algebra, i.e. Hochschild cohomology (by taking the derived $(\infty,1)$-category of bimodules over an associative algebra), and a notion of derived centre of a monoidal $(\infty,1)$-category, i.e. the derived Drinfeld centre (by taking the $(\infty,1)$-category of (presentable) $(\infty,1)$-categories).

Answer 2

The key formula is, as you note, $End_{A\text{-mod-}A} (A)$. We can then interpret this same formula in lots of settings.

In the category of sets, $A$ is an ordinary algebra, and $End_{A\text{-mod-}A} (A)$ is the ordinary center of A. (The only things that commute with the left action are right multiplication by elements, and those only commute with right multiplication if the element is central.)

In the derived setting you get the derived version of the center, namely Hochschild cohomology.

In the $(2,1)$-category of categories, an associative algebra is exactly a monoidal category, and $End_{A\text{-mod-}A} (A)$ is the Drinfel'd center. To see this you need to be a bit careful unpacking what a functor of bimodule categories means. In particular instead of a condition saying $f(am) = af(m)$ there's a natural isomorphism $f(a \otimes m) \rightarrow a f(m)$ satisfying a coherence condition (see Ostrik for more details). Again left $A$-module endofunctors of $A$ correspond exactly to tensoring on the right with some object of $A$ (with the associator for $A$ giving the natural transformation). In order for tensoring on the right with an object to commute with the right action, we need to pick isomorphisms $\eta_b: a \otimes b \rightarrow b \otimes a$. In other words, $End_{A\text{-mod-}A} (A)$ consists of objects together with half-braidings and so is the Drinfeld center.