A Hovey triple arising from two cotorsion pairs

TL;DR

This paper constructs a unique Hovey triple from two hereditary cotorsion pairs in an extriangulated category satisfying Condition (WIC), generalizing previous results and revealing new phenomena in triangulated categories.

Contribution

It introduces a method to derive a Hovey triple from two hereditary cotorsion pairs under specific conditions, extending Gillespie's work to a broader context.

Findings

Existence of a unique thick class forming a Hovey triple

Generalization of Gillespie's results to extriangulated categories

New phenomena observed in triangulated categories

Abstract

Assume that $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$ is an extriangulated category satisfying Condition (WIC). Let $(\cal Q, \widetilde{\cal R})$ and $(\widetilde{\cal Q}, \cal R)$ be two hereditary cotorsion pairs satisfying conditions $\widetilde{\cal R} \subseteq \cal R$, $\widetilde{\cal Q}\subseteq \cal Q$ and $\widetilde{\cal Q}\cap \cal R = \cal Q \cap \widetilde{\cal R}$. Then there exists a unique thick class $\cal W$ for which $(\cal Q,\cal W,\cal R)$ is a Hovey triple. As an application, this result generalizes the work by Gillespie in an exact case. Moreover, it highlights new phenomena when it applied to triangulated categories.