$G$-symmetric monoidal categories of modules over equivariant commutative ring spectra

Abstract

We describe the multiplicative structures that arise on categories of equivariant modules over certain equivariant commutative ring spectra. Building on our previous work on ${\mathcal{N}}_{\infty }$ ring spectra, we construct categories of equivariant operadic modules over ${\mathcal{N}}_{\infty }$ rings that are structured by equivariant linear isometries operads. These categories of modules are endowed with equivariant symmetric monoidal structures, which amounts to the structure of an “incomplete Mackey functor in homotopical categories”. In particular, we construct internal norms which satisfy the double coset formula. One application of the work of this paper is to provide a context in which to describe the behavior of Bousfield localization of equivariant commutative rings. We regard the work of this paper as a first step towards equivariant derived algebraic geometry.