Comodules and coproduct decomposition

Ross Street – 21 September 2005

This talk was motivated by work with Brian Day on the centre (and lax centre) of a monoidal category. We simply wanted a proof, with suitable conditions on a monoidal category A, that the centre of the category AG of representations of a finite group G in a category A is equivalent to the category AG' of functors from G' to A (where G' is the groupoid whose objects are elements of G and whose morphisms are conjugators). The proof should apply to A = Set and A = Vect. The approach was to remind the audience of Yetter-Drinfeld modules (also called crossed bimodules) for a bimonoid H in a symmetric monoidal A. In particular, such a YD-module is an H-comodule. In the case where H is the coproduct of G copies of the tensor unit in A (that is, G itself when A = Set and the group algebra of G when A = Vect), we examined H-comodules showing that for very general A they have a coproduct decomposition indexed by G. This situation was studied in detail by L. Coppey and Kelly-Pultr in the 1970s however their emphasis seemed more on the dual question of product decomposition. The question concerns the comonadicity of the functor "coproduct" from the product of G copies of A to A. Day-Lack-Street have some unpublished results about when coreflexive equalizers are always split which, together with the requirement that the coproduct coprojections be strong monomorphisms, certainly imply the comonadicity. Armed with the coproduct decomposition it is an easy matter to use the other conditions on a YD-module to turn it into a functor G' --> A.