Abstract

When G is a finite group, rational 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 of rational G-Mackey functors splits as a product of module categories with Weyl group actions.  Classically, the homotopy theory of G-spectra depended on a choice of a universe – this was modelled, for example, by the orthogonal G-spectra of Mandell and May. In modern language, Blumberg and Hill generalised the construction of the homotopy theory of G-spectra and show that one can choose an additive structure of it to be modelled by a homotopy type of an N_\infty operad O (or equivalently a transfer system associated to O). Thus homotopy groups of a G-spectrum with additive structure given by O have those additive transfers that are parametrised by O. We call such a structure a G-Mackey functor for O. The purpose of this talk is to describe splittings of rational G-Mackey functors for incomplete transfer systems and discuss how this extends to the topological setting. This is joint work with Anna Marie Bohmann, Dave Barnes and Mike Hill.