A General Approach to Green Functors Using Bisets

We introduce a biset-theoretic notion of a Green functor which accommodates the functorial and ring-theoretic structural features of the modular character functor. For any Green functor A in that sense, we introduce an algebra (A). Any (A)-module has the same kind of functorial structure as A and is also a module for the algebra , where G runs over the underlying family of finite groups. In a globally defined scenario and also in a scenario localized to the subquotients of a fixed finite group, we take A to be the modular character functor, and we classify the simple (A)-modules.