# What is known about module categories over general monoidal categories?

All of the literature I have seen on module categories over monoidal categories has been in the rigid $k$-linear semisimple case, more or less in the spirit of Ostrik's paper, Module categories, weak Hopf algebras and modular invariants. However, the basic definition of module category makes perfect sense in general, so I am wondering if there has been any work done in more general settings.

In particular, I am interested in the following sort of question. Let $\mathcal{C}$ be a braided monoidal category (maybe ultimately with some added assumptions, such as rigidity). Any left $\mathcal{C}$-module category is automatically a $(\mathcal{C}, \mathcal{C})$-bimodule category via the braiding. For bimodule categories there should be a good notion of "tensoring over $\mathcal{C}$," which makes the 2-category of bimodules (and hence the 2-category of left modules) into a monoidal 2-category. If we extract invertible objects and morphisms at all levels, we should obtain a sort of Brauer 3-group for $\mathcal{C}$. (In the fusion category setting, this object is considered in the recent preprint by Etingof, Nikshych, and Ostrik, Fusion categories and homotopy theory.)

I'd like to know how this is related to the "internal" Brauer 3-group of $\mathcal{C}$, whose objects are Azumaya algebras in $\mathcal{C}$, morphisms are invertible bimodules, and 2-morphisms are invertible bimodule morphisms. In the fusion category setting, the connection is furnished by the main theorem in Ostrik's paper, which states that any semisimple indecomposable module category is equivalent to the category of modules over some algebra in $\mathcal{C}$. (I think this implies that the two notions of Brauer 3-group are equivalent in the fusion category setting.) Is any result along these lines known in more generality? It's not even clear to me that the module category of left modules over an Azumaya algebra in $\mathcal{C}$ is invertible, or even what the tensor product of two module categories of modules is.

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Could you recall the definition of Azumaya algebra in a general braided monoidal category? –  Reid Barton Nov 25 '09 at 7:23
An Azumaya algebra is one in which the tensor actions $X \mapsto A \otimes X$ and $X \mapsto X \otimes A$ are equivalences between $\mathcal{C}$ and the categories of left $A \otimes A^{\text{op}}$-modules and right $A^{\text{op}} \otimes A$-modules, respectively. This definition comes from the paper by Van Oystaeyen and Zhang, "The Brauer Group of a Braided Monoidal Category." –  Evan Jenkins Nov 25 '09 at 18:09

In Monoidal 2-structure of Bimodule Categories Greenough discusses some of what you seem to be asking here for arbitrary abelian categories. He describes the tensor product $\mathcal{M}\boxtimes_{\mathcal{C}}\mathcal{N}$ of $\mathcal{C}$-bimodule categories $\mathcal{M},\mathcal{N}$ over a tensor category $\mathcal{C}$ as universal with respect to "$\mathcal{C}$-balanced" functors on the Deligne product $\mathcal{M}\boxtimes\mathcal{N}$.