The double of a Hopf monad
  The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of 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 of T over Z_T. By Beck's theory, this has two consequences. On one hand, D_T=Z_T 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 also lifts Z_T to a Hopf monad on T-C which gives the coend of T-C. Hence, for T=Z, an explicit description of the Hopf algebra structure of the coend of Z(C) in terms of the structural morphisms of C, which is useful in quantum topology. 
Alain Brugui\`eres and Alexis Virelizier