Quantum double of Hopf monads and categorical centers

Abstract

The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of a Hopf monad naturally arises if one tries to reconstruct the structure of Z(C) in terms of its monad Z: we show that Z is a quasitriangular Hopf monad on C and Z(C) is isomorphic to the braided category Z − C of Z-modules. More generally, let T be a Hopf monad on an autonomous category C. We construct a Hopf monad Z T on C, the centralizer of T , and a canonical distributive law Ω : T Z T → Z T T. By Beck's theory, this has two consequences. On one hand, D T = Z T • Ω T is a quasitriangular Hopf monad on C, called the double of T , and Z(T − C) D T − C as braided categories. As an illustration, we define the double D(A) of a Hopf algebra A in a braided autonomous category in such a way that the center of the category of A-modules is the braided category of D(A)-modules (generalizing the Drinfeld double). On the other hand, the canonical distributive law Ω also lifts Z T to a Hopf monad˜Zmonad˜ monad˜Z Ω T on T −C, and˜Zand˜ and˜Z Ω T (, T 0) is the coend of T −C. For T = Z, this gives an explicit description of the Hopf algebra structure of the coend of Z(C) in terms of the structural morphisms of C. Such a description is useful in quantum topology, especially when C is a spherical fusion category, as Z(C) is then modular.