# 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,

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.

• Could you recall the definition of Azumaya algebra in a general braided monoidal category? Nov 25, 2009 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." Nov 25, 2009 at 18:09
• The Deligne tensor product doesn't exist for arbitrary abelian categories. But if you relax "abelian" to instead mean "has finite colimits" then there is a very general tensor product that always exists called the "Kelly tensor product". A similar statement for locally presentable categories is in Cor 2.2.5 of (arXiv:1105.3104). I think the categorical machinery underlying these results can be adapted to the case of tensor products of module cats in the generality you are considering, but I don't know of a reference that does it. Jul 27, 2017 at 19:47
• The "module categories are categories of modules" yoga is probably very special to the finite rigid setting. We tried to be careful about what exactly was needed for this here:arxiv:1406.4204. An interesting example is the following (non-rigid) tensor category C. It is finite, semisimple and has two simple objects 1 and x. The object 1 is the unit and we have $x \otimes x = 0$. This determines the monoidal structure by linearity. If I remember right, the subcategory generated by x is a module category which is not of the form A-mod(C) for an algebra A in C. Jul 27, 2017 at 19:57