Let $\mathcal C$ be a monoidal category. Recall that the *(Drinfel'd) center* of $\mathcal C$ is the braided monoidal category $Z(\mathcal C)$ with:

- Objects: pairs $M \in \mathcal C$ and $\mu: M\otimes(-) \overset\sim\to (-)\otimes M$ a natural iso, which is required to satisfy: $\mu: M \otimes \mathbb 1 \to \mathbb 1 \otimes X$ is the canonical isomorphism $X\otimes 1 \to M \to 1\otimes M$, and $\mu_{A\otimes B} = (\operatorname{id}_A \otimes \mu_B)\circ (\mu_A\otimes \operatorname{id}_B)$ as maps $M\otimes A \otimes B \to A\otimes B \otimes M$. (I will drop all associators.) So $\mu$ is tryin to be a "braiding" — the second axiom is some sort of "hexagon" axiom.
- Morphisms: a map $(M,\mu) \to (N,\nu)$ is a morphism $f: M \to N$ so that $\nu \circ (f\otimes \operatorname{id}) = (\operatorname{id}\otimes f) \circ \mu$ as natural transformations $M\otimes (-) \to (-) \otimes N$. (Here $f\otimes \operatorname{id}$ is the natural transformation $M\otimes \to N\otimes$, etc.)
- The monoidal structure is $(M,\mu)\otimes (N,\nu) = (M\otimes N, (\mu \otimes \operatorname{id}_N) \circ (\operatorname{id}_M \otimes \nu))$.
- The braiding $(M,\mu) \otimes (N,\nu) \to (N,\nu)\otimes (M,\mu)$ is given my $\mu_N : M\otimes N \to N\otimes M$. It is a fun calculation to check that this is a (bi-natural) isomorphism in the category, and satisfies all requirements to be a braiding.

An important example: when $\mathcal C$ is the representation theory of a finite abelian group $G$, and writing $G^\vee$ for the Pontrjagin dual to $G$ (it is isomorphic for finite abelian groups), then as a monoidal category, *but not as a braided monoidal category*, $Z(\mathcal C)$ is (equivalent to) the representation theory of $G\times G^\vee$.

When $\mathcal C$ is braided (with braiding $\beta_{M,N} : M\otimes N \to N\otimes M$), there is a full, faithful injection $\mathcal C \hookrightarrow Z(\mathcal C)$, which is usually not an essential surjection, given on objects by $M \mapsto (M,\beta_{M,-})$. (For the finite abelian group case, this map is dual to the projection $G\times G^\vee \to G$.)

In addition to representations of monoids, another important source of categories are as categories of modules of commutative rings. I think I can prove that if $R$ is a commutative ring, then the injection $\operatorname{Mod}(R) \to Z(\operatorname{Mod}(R))$ is an equivalents of categories (the monoidal structure on $\operatorname{Mod}(R)$ is $\otimes_R$). (Although I've read that the center of $\operatorname{Mod}(R)$ is supposed to be more like sheaves on the loop space of $\operatorname{Spec}(R)$, so I might have made an error.)

My question is about the converse.

The weak statement I do not expect to be true:

Question (weak assumptions):Suppose that $(\mathcal C,\otimes)$ is a monoidal category (abelian, complete, cocomplete, etc. ... maybe also some statement along the "tensor products distribute over filtered limits" variety) and that there exists a monoidal equivalence $\mathcal C \to Z(\mathcal C)$. Must $(\mathcal C,\otimes)$ be equivalent to $(\operatorname{Mod}(R),\otimes_R)$ for some commutative ring $R$?

A more reasonable question is:

Question (stronger assumptions):Suppose that $(\mathcal C,\otimes,\beta)$ is abraidedmonoidal category (with more adjective?) such that the canonical map $\mathcal C \to Z(\mathcal C)$ is an equivalence of braided monoidal categories. Must $(\mathcal C,\otimes,\beta)$ be equivalent as a braided category to $(\operatorname{Mod}(R),\otimes_R,\operatorname{flip})$ for some commutative ring $R$?

Probably (super)commutative rings in SuperVectorSpaces will provide counterexamples. And so the real question is:

Real question:What is / is there a nice characterization of those monoidal categories that are equivalent to their centers?

is notsymmetric, but just braided, so I don't really know how much geometry to hope for.) $\endgroup$ – Theo Johnson-Freyd Oct 22 '10 at 3:51