Tony Zorman

Let B be a bialgebra and let C be the category of finite-dimensional B-comodules. We extend the reconstruction theory for C-module categories from the case where B is Hopf to the general setting, replacing algebra objects by lax module functors and monads. 

Crucially, lax C-module endofunctors turn out to correspond to "Hopf trimodules"—Hopf modules equipped with two coactions and one action. 

Using this, we show that any (locally) finite abelian C-module category is of the form A-(co)mod, where A is a Hopf trimodule (finite dual co)algebra. This allows us to give conceptual proofs of the Hausser–Nill theorem and the fundamental theorem of Hopf modules. 

This is joint work in progress with Matti Stroiński.