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

TL;DR

This paper develops the theory of equivariant modules over N-infinity ring spectra, establishing their symmetric monoidal structures and internal norms, advancing the foundation for equivariant derived algebraic geometry.

Contribution

It constructs categories of equivariant operadic modules with symmetric monoidal structures and internal norms, extending previous work on N-infinity ring spectra.

Findings

Categories of equivariant modules have equivariant symmetric monoidal structures.

Internal norms satisfy the double coset formula.

Framework supports future development of equivariant derived algebraic geometry.

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 N-infinity ring spectra, we construct categories of equivariant operadic modules over N-infinity 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. We regard the work of this paper as a first step towards equivariant derived algebraic geometry.