Biset functors for categories

Abstract

We introduce the theory of biset functors defined on finite categories. Previously, biset functors have been defined on groups, and in that context they are closely related to Mackey functors. Standard examples on groups include representation rings, the Burnside ring and group cohomology. The new theory allows these same examples, but defined for arbitrary finite categories, thus including, for instance, the representation rings of finite EI categories, among which are posets, and the free categories associated to quivers without oriented cycles. Group homology with trivial coefficients is replaced by the homology of the category with constant coefficients and this is a biset functor when bisets that are representable on one side are used. We give a definition of the Burnside ring of an arbitrary finite category. It is a biset functor that plays a key role throughout the theory. We discuss properties of the simple biset functors on categories, including their parametrization and calculation. We describe the symmetric monoidal structure on biset functors, Green biset functors and an approach to fibered biset functors for categories, together with the technicalities these entail. Various examples are given, the most elaborate showing a connection between the correspondence functors of Bouc and Th\'evenaz and biset functors on Boolean lattices.