Model Structures Arising from Extendable Cotorsion Pairs

TL;DR

This paper develops new model structures from extendable cotorsion pairs in exact categories, leading to novel descriptions of derived categories and interpretations of recollements in higher dimensions.

Contribution

It introduces methods to construct hereditary Hovey triples from extendable cotorsion pairs, unifying various higher-dimensional homotopy theories.

Findings

New description of $Q$-shaped derived categories

Interpretation of Krause's recollement in $n$-dimensional homotopy categories

Two approaches to $n$-dimensional hereditary Hovey triples that coincide

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 $\mathcal{A}$.