Eilenberg-moore Model Categories and Bousfield Localization

Talk 1: Big Goal of Alg Top, operads and model categories, fix notation for model categories, remarks about how difficult it is to verify model category axioms. Motivation from equivariant spectra, and discussion of Kervaire. Monoidal model categories, define the inherited model structure on the category of algebras over an operad. Basic facts about Bousfield localization. Preservation theorem and proof, a word about semi-model categories. Talk 2: Preliminary results about why semi-model categories are not so bad. General transfer principles for putting (semi) model structures on T-algebras. Review of Schwede-Shipley proof for monoids. Connection to tame polynomial monads. Version for commutative monoids. Examples of model categories satisfying all our axioms so far. Notion of cofibrancy for operads. John Harper’s filtration and resulting axioms to get semi-model structures on algebras over operads. Examples of localization preserving structure: truncation in spaces, homological algebra, stable localizations in spectra, bringing it back to the example of equivariant spectra.