Left-induced model structures and diagram categories

TL;DR

This paper establishes existence results for left-induced model structures, explores fibrant generation and Postnikov presentations, and applies these concepts to diagram categories, differential graded Hopf algebras, and Reedy categories.

Contribution

It develops new existence theorems for left-induced model structures and extends fibrant generation and Postnikov presentation concepts to broader contexts.

Findings

Established existence results for left-induced model structures.

Constructed a left-induced model structure on augmented H-comodule algebras.

Showed the category of bounded below chain complexes has a Postnikov presentation.

Abstract

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.