Tony Zorman

Let C be a monoidal category and M a C-module category. 

There are many results specifying finiteness, exactness, and rigidity/closedness conditions on C and M so that M can be realized as the category of modules for an algebra object A. Here, A can be an object of C, or a larger monoidal category, such as Ind(C). 

We show that the category Tamb(C) of Tambara modules on C is the universal category for this purpose: algebra objects in Tamb(C) describe all C-module categories, with no assumptions on C or M. Using the structure of the category Tamb(D-proj) for a finite tensor category D, we verify a conjecture of Etingof-Ostrik on a generalization of a theorem of Skryabin: the category of modules for an algebra object A in D is an exact module category if and only if A is semisimple. 

Additionally, we obtain an analogue of the Jacobson radical for algebra objects in D. 

This is joint work in progress with Mateusz Stroiński