Barker, L.2018-04-122018-04-1220160092-7872http://hdl.handle.net/11693/36633We 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 ΛA. Any ΛA-module has the same kind of functorial structure as A and is also a module for the algebra ⊕G A(G) , 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 ΛA-modules. © 2016, Copyright © Taylor & Francis Group, LLC.EnglishGreen biset functorGreen categoryModular character functorSynthetic algebraPrimary: 20C20Secondary: 19A22Article10.1080/00927872.2016.11726011532-4125