Hopf monads on monoidal categories, and modularity of the centre

Alain Bruguières – 10 December 2008

The talk comes in two independent parts. In the first part, I will explain how, with Steve Lack's help, we could at a single stroke simplify the definition of Hopf monad, and extend it to arbtrary monoidal categories, and simplify the proof of the multiplicativity of the antipodes. I will also explain how this new definition can be interpreted in the case of a monoidal closed category, where being Hopf can be readily encoded in terms of 2-variable antipodes (recall that in the autonomous case we encoded it in terms of 1-variable antipodes). In the second part, I will recall the construction of the double of a Hopf monad on an autonomous category, which is based on the canonical lift of centralizers. I will apply this lifting property to fusion categories, and explain how this allows one to prove the modularity of the center of an arbitrary spherical fusion category over a ring (generalizing Mueger's modularity theorem, which concerns spherical fusion categories of invertible dimension over an algebraically closed field).