Mackey group categories and their simple functors

Abstract

Constructing the Mackey group category M using axioms which are reminiscent of fusion systems, the simple RM-functors (the simple functors from the R-linear extension of M to R-modules, where R is a commutative ring) can be classified via pairs consisting of the objects of the Mackey group category (which are finite groups) and simple modules of specific group algebras. The key ingredient to this classification is a bijection between some RM-functors (not necessarily simple) and some morphisms of EndRM(G). It is also possible to define the Mackey group category by using Brauer pairs, or even pointed groups as objects so that this classification will still be valid.