Model Structures Arising from Extendable Cotorsion Pairs

Abstract

The aim of this paper is to construct exact model structures from so called extendable cotorsion pairs. Given a hereditary Hovey triple $(\mathcal{C}, \mathcal{W}, \mathcal{F})$ in a weakly idempotent complete exact category with enough projectives and injectives. If one of the cotorsion pairs $(\mathcal{C}\cap\mathcal{W}, \mathcal{F})$ and $(\mathcal{C}, \mathcal{W}\cap\mathcal{F})$ is extendable, then there is a chain of hereditary Hovey triples whose corresponding homotopy categories coincide. As applications, we obtain a new description of the $Q$-shaped derived categories introduced by Holm and J\o rgensen. We can also interpret the Krause's recollement in terms of ``$n$-dimensional'' homotopy categories. Finally, we have two approaches to get ``$n$-dimensional'' hereditary Hovey triples, which are proved to coincide, in the category Rep$(Q,\mathcal{A})$ of all representations of a rooted quiver $Q$ with values in an abelian category $\mathcal{A}$.