Left-induced model structures and diagram categories
  We prove existence results a la Jeff Smith for left-induced model category structures, of which the injective model structure on a diagram category is an important example. We further develop the notions of fibrant generation and Postnikov presentation from Hess, which are dual to a weak form of cofibrant generation and cellular presentation. As examples, for k a field and H a differential graded Hopf algebra over k, we produce a left-induced model structure on augmented H-comodule algebras and show that the category of bounded below chain complexes of finite-dimensional k-vector spaces has a Postnikov presentation.   To conclude, we investigate the fibrant generation of (generalized) Reedy categories. In passing, we also consider cofibrant generation, cellular presentation, and the small object argument for Reedy diagrams. 
Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kedziorek,   Emily Riehl, and Brooke Shipley
math.AT math.CT