Equivariant symmetric monoidal structures

TL;DR

This paper introduces $G$-symmetric monoidal categories as an equivariant extension of symmetric monoidal categories, enabling the study of $G$-commutative monoids and their behavior under Bousfield localization in equivariant spectra.

Contribution

It defines $G$-symmetric monoidal categories and $G$-commutative monoids, extending classical structures to the equivariant setting and analyzing their stability under localization.

Findings

Bousfield localization preserves $G$-commutative monoids under certain conditions

New framework for equivariant algebraic structures in homotopy theory

Applications to categories of modules over $G$-commutative monoids

Abstract

Building on structure observed in equivariant homotopy theory, we define an equivariant generalization of a symmetric monoidal category: a $G$-symmetric monoidal category. These record not only the symmetric monoidal products but also symmetric monoidal powers indexed by arbitrary finite $G$-sets. We then define $G$-commutative monoids to be the natural extension of ordinary commutative monoids to this new context. Using this machinery, we then describe when Bousfield localization in equivariant spectra preserves certain operadic algebra structures, and we explore the consequences of our definitions for categories of modules over a $G$-commutative monoid.