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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
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