Rigidity in equivariant stable homotopy theory

Abstract

For any finite group $G$, we show that the $2$–local $G$–equivariant stable homotopy category, indexed on a complete $G$–universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy cofiber sequences and the stable Burnside category determine all “higher-order structure” of the $2$–local $G$–equivariant stable homotopy category, such as the equivariant homotopy types of function $G$–spaces. Our result can be seen as an equivariant version of Schwede’s rigidity theorem at the prime $2$.