Generalized Green functors and semisimplicity

Abstract

A pursuit of abstract generality: the theory of biset functors provides a framework for the globalization of Mackey functors. In this setting, linear morphisms between two finite groups are indexed by conjugacy classes of subgroups of their direct product. Although this formalism has proved useful in many situations, there exist Mackey functors that do not admit a global description within the theory of biset functors. Restricting attention to Green biset functors, and taking as a model an object introduced by Boltje and Danz, we introduce a generalization, what we call the theory of Green prebiset functors. In this extended setting, linear morphisms between two finite groups are indexed by all subgroups of the direct product. The conjugacy classes then arise as orbit sums with respect to the conjugation action. In a similar way, we obtain Green biset functors as special cases of Green prebiset functors. The results obtained in our framework are partial and are discussed in the introduction.