Cut notions in extriangulated categories related to Auslander-Buchweitz theory and cotorsion theory

TL;DR

This paper extends Auslander-Buchweitz and cotorsion theories to extriangulated categories by considering subcategories, broadening their applicability beyond abelian, exact, and triangulated categories.

Contribution

It introduces relative notions in extriangulated categories using subcategories, unifying and extending existing theories in various categorical contexts.

Findings

Develops a framework for relative Auslander-Buchweitz theory in extriangulated categories.

Establishes conditions under which cotorsion pairs can be constructed in this setting.

Provides a unified approach that encompasses abelian, exact, and triangulated categories.

Abstract

In this work we introduce notions in Auslander-Buchweitz theory and cotorsion theory in extriangulated categories which extend the given ones for abelian categories. Although these notions have been already developed for extriangulated categories in remarkable works, in this paper, we tackle them in a relative sense by considering subcategories of objects. This approach not only covers the existing theory given on abelian, exact and triangulated categories, but it also shows how to get similar results with an appropriate treatment of local properties.