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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeMay 18th 2012

The preprint The homotopy theory of coalgebras over a comonad by Hess and Shipley looks interesting. Among other things it contains

• a theorem about replacing a model structure by a Quillen equivalent one in which the cofibrations are the monomorphisms
• a theorem about when the category of coalgebras for a comonad inherits a model structure by “left transfer”.

I wonder whether there could be a model structure on coalgebraically cofibrant objects?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 18th 2012

Ah, thanks for pointing this out. I had heard them speak about that, but didn’t see the article appear.

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 9th 2014

The “left transfer” part is developed further in this paper: http://arxiv.org/abs/1401.3651, which basically develops a theory dual to the usual theory of transferred model structures:

Left-induced model structures and diagram categories

Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kedziorek, Emily Riehl, Brooke Shipley

We prove existence results for and verify certain elementary properties of left-induced model structures, of which the injective model structure on a diagram category is an important example. We refine our existence results and prove additional properties for the injective model structure. To conclude, we investigate the fibrant generation of (generalized) Reedy categories. In passing, we also consider the cofibrant generation, cellular presentation, and small object argument for Reedy diagrams.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 9th 2014

Thanks. I have added that citation now to the References-section at transferred model structure

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 5th 2016

Just for completeness, I gave the article a home at model structure on coalgebras over a comonad (to go along with model structure on algebras over a monad)

Currently this is nothing but a stub.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 5th 2016

Plenty of technical conditions there. But I suppose if we start with the local injective model structure on simplicial presheaves over formal smooth manifolds, and fix any object $\Sigma$, then the jet comonad $J^\infty_\Sigma$ will satisfy the conditions of theorem 5.8?

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 7th 2016

To construct a cotransferred model structure along a left adjoint functor that lands in a combinatorial model category, the only condition to check is that maps with a right lifting property with respect to newly created cofibrations are (newly created) weak equivalences.

This is actually an immediate consequence of the Smith recognition theorem (so the above cited result of 6 authors is mostly of expository nature), combined with a recent result by Makkai and Rosický that shows the resulting model structure to be cofibrantly generated.