Bousfield Localization and Algebras over Colored Operads

Abstract

We provide a very general approach to placing model structures and semi-model structures on algebras over symmetric colored operads. Our results require minimal hypotheses on the underlying model category $\mathcal {M}$ , and these hypotheses vary depending on what is known about the colored operads in question. We obtain results for the classes of colored operad which are cofibrant as a symmetric collection, entrywise cofibrant, or arbitrary. As the hypothesis on the operad is weakened, the hypotheses on $\mathcal {M}$ must be strengthened. Via a careful development of the categorical algebra of colored operads we provide a unified framework which allows us to build (semi-)model structures for all three of these classes of colored operads. We then apply these results to provide conditions on $\mathcal {M}$ , on the colored operad O, and on a class $\mathcal {C}$ of morphisms in $\mathcal {M}$ so that the left Bousfield localization of $\mathcal {M}$ with respect to $\mathcal {C}$ preserves O-algebras. Even the strongest version of our hypotheses on $\mathcal {M}$ is satisfied for model structures on simplicial sets, chain complexes over a field of characteristic zero, and symmetric spectra. We obtain results in these settings allowing us to place model structures on algebras over any colored operad, and to conclude that monoidal Bousfield localizations preserve such algebras.