Abstract

Rationally, genuine G-spectra split as a product of simpler categories: rational spectra with Weyl group actions. Algebraically, this is the statement that rational Mackey functors are semi-simple and the category splits as a product of module categories with Weyl group actions. A current and developing research theme is incomplete stable equivariant homotopy theory. Classically this meant using G-spectra with an incomplete universe. In modern language, incomplete stable equivariant homotopy theory means using a transfer system that is not complete. The translation between these two viewpoints is via N-infinity operads, considered as giving the additive structure to G-spectra (rather than then multiplicative structure).  With that heavyweight motivation given, the purpose of this talk is to describe splittings of rational Mackey functors for incomplete transfer systems and discuss how this may extend to the topological setting. This is joint work with Anna Marie Bohmann, Magdelena Kedziorek and Mike Hill.