  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.
