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,

- Ostrik, V.
*Module categories, weak Hopf algebras and modular invariants*, Transformation Groups**8**(2003) 177–206, doi:10.1007/s00031-003-0515-6, arXiv:math/0111139.

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

- Pavel Etingof, Dmitri Nikshych, Victor Ostrik,
*Fusion categories and homotopy theory*, Quantum Topol.**1**(2010), 209–273, journal, arXiv:0909.3140.

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.