Equivariant symmetric monoidal structures

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.

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