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 : TZ_T→Z_TT. By Beck's theory, this has two consequences. On one hand, D_T Z_T oΩ T is a quasitriangular Hopf monad on C, called the double of T, and Z(T — C) DT — 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 Z~Ω_T on T—C, and Z~Ω_T(1, T_o) 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.
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