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The preprint The homotopy theory of coalgebras over a comonad by Hess and Shipley looks interesting. Among other things it contains
I wonder whether there could be a model structure on coalgebraically cofibrant objects?
Ah, thanks for pointing this out. I had heard them speak about that, but didn’t see the article appear.
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.
Thanks. I have added that citation now to the References-section at transferred model structure
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.
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?
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.
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