Model Structures Arising from Extendable Cotorsion Pairs
Qingyu Shao, Junpeng Wang, Xiaoxiang Zhang
TL;DR
The paper develops a framework to build exact model structures from extendable cotorsion pairs within hereditary Hovey triples on weakly idempotent complete exact categories. It proves that if a cotorsion pair at the heart of a Hovey triple is left extendable, one obtains an infinite, compatible chain of hereditary Hovey triples with identical homotopy categories, enabling new descriptions of the unbounded derived category $\mathbf{D}(R)$ and a recollement interpretation via $n$-dimensional homotopy categories. The results extend to representation categories $\mathrm{Rep}(Q,\mathcal{A})$, showing two constructions of $n$-dimensional hereditary Hovey triples coincide, and provide a unified view linking cotorsion-pair theory, model structures, and derived/recollement phenomena. Collectively, these findings offer a versatile, categorical toolbox for understanding derived categories, recollements, and model structures in module and quiver-representation settings.
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. 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 unbounded derived category $\mathbf{D}(R)$ over a ring $R$. Moreover, we can 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 $\mathcal{A}$.