How to construct a Hovey triple from two cotorsion pairs

TL;DR

This paper provides a method to construct a Hovey triple, which is an abelian model structure, from two compatible complete hereditary cotorsion pairs in an abelian or exact category, unifying and extending existing frameworks.

Contribution

The authors introduce a novel construction of a Hovey triple from two specific cotorsion pairs, establishing conditions for a unique abelian model structure.

Findings

Constructs a unique abelian model structure from two cotorsion pairs.

Defines conditions for the classes of cofibrant and fibrant objects.

Extends the theory of model structures in abelian categories.

Abstract

Let $\mathcal{A}$ be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(\mathcal{Q}, \widetilde{\mathcal{R}})$ and $(\widetilde{\mathcal{Q}}, \mathcal{R})$ in $\mathcal{A}$ satisfying $\widetilde{\mathcal{R}} \subseteq \mathcal{R}$ and $\mathcal{Q} \cap \widetilde{\mathcal{R}} = \widetilde{\mathcal{Q}} \cap \mathcal{R}$. We show how to construct a (necessarily unique) abelian model structure on $\mathcal{A}$ with $\mathcal{Q}$ (respectively $\widetilde{\mathcal{Q}}$) as the class of cofibrant (resp. trivially cofibrant) objects and $\mathcal{R}$ (respectively $\widetilde{\mathcal{R}}$) as the class of fibrant (resp. trivially fibrant) objects.