Bousfield Localization and Algebras over Operads

We give conditions on a monoidal model category M and on a set of maps S so that the Bousfield localization of M with respect to S preserves the structure of algebras over various operads. This problem was motivated by an example due to Mike Hill which demonstrates that for the model category of equivariant spectra, even very nice localizations can fail to preserve commutativity. As a special case of our general machinery we characterize which localizations preserve genuine equivariant commutativity. Our results are general enough to hold for non-cofibrant operads as well, and we will demonstrate this via a treatment of when localization preserve strict commutative monoids. En route we will introduce the commutative monoid axiom, which guarantees us that commutative monoids inherit a model structure. If there is time we will say a word about the generalizations of this axiom to other non-cofibrant operads, and about how these generalized axioms interact with Bousfield localization.