In general, it is difficult to transfer a model structure from a monoidal
model category M to the category of algebras over a (colored) operad P in M.
Often, one only ends up with a semi-model structure (i.e. where half the
factorization and lifting axioms only hold for maps with cofibrant domain),
and even this traditionally requires P to be Sigma-cofibrant. I?ll explain
what standard techniques still work in the context of semi-model categories,
so that users of model categories can get by if they only have a semi-model
structure. I?ll then report on recent joint work with Donald Yau providing
model structures and semi-model structures for algebras over non-cofibrant
colored operads. As an application, we?ll prove a very general result
regarding preservation of algebras by left and right Bousfield
localizations. Examples will include spaces, spectra, equivariant spaces and
spectra, categories, chain complexes, and the stable module category.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/6449