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Mark A. Hovey






We give conditions on a monoidal model category M and on a set of maps C so that the Bousfield localization of M with respect to C preserves the structure of algebras over various operads. This problem was motivated by a step in the solution of the Kervaire Invariant One Theorem and by an example which demonstrates that for the model category of equivariant spectra, preservation does not come for free, even for cofibrant operads. We provide a general theorem regarding when localization preserves P-algebra structure for an arbitrary operad P. We then characterize the localizations which respect monoidal structure and prove that all such localizations preserve algebras over cofibrant operads. As a special case we recover numerous classical theorems about preservation of algebraic structure under localization, we improve upon known results regarding preservation for equivariant spectra, and we introduce a collection of operads which allow us to study the phenomenon of localization destroying some, but not all, of equivariant commutative structure.

To demonstrate our preservation result for non-cofibrant operads, we develop a theory for when the category of commutative monoids in M inherits a model structure from M in which a map is a weak equivalence or fibration if and only if it is so in M. We then investigate properties of cofibrations of commutative monoids, functoriality of the passage from a commutative monoid R to the category of commutative R-algebras, rectification between E-algebras and commutative monoids, and the relationship between commutative monoids and monoidal Bousfield localization functors. We recover numerous known examples and a few new examples of model categories in which commutative monoids inherit a model structure. We then work out when localization preserves commutative monoids and the commutative monoid axiom. Finally, we provide conditions so that a left Bousfield localization satisfies the monoid axiom.



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