Chains of model structures arising from cotorsion pairs on extriangulated categories

TL;DR

This paper explores chains of model structures derived from cotorsion pairs in extriangulated categories, establishing equivalences of their homotopy categories and applying these to Gorenstein injective objects.

Contribution

It constructs new hereditary Hovey triples from existing ones, refining previous results and providing new examples in derived categories under mild assumptions.

Findings

Homotopy categories are equivalent under certain conditions.

Chains of model structures can be constructed with triangulated equivalent homotopy categories.

Applications include Gorenstein injective modules and cohomological ghost triangles.

Abstract

The main aim of this paper is to study chains of model structures arising from cotorsion pairs in extriangulated categories. Starting with a hereditary Hovey triple, we construct further hereditary Hovey triples whose homotopy categories are equivalent under suitable completeness assumptions, thereby refining results due to El Maaouy and Shao-Wang-Zhang. As an application, we consider objects of finite Gorenstein injective dimension with respect to a proper class of $\mathbb{E}$-triangles. Under mild set-theoretic assumptions, we obtain a chain of model structures whose homotopy categories are all triangulated equivalent to a common stable category. This recovers known results for Gorenstein injective modules and yields new examples in the derived category of a ring when the proper class is given by cohomological ghost triangles.