Tilting subcategories in extriangulated categories

TL;DR

This paper introduces a new notion of tilting subcategories in extriangulated categories, providing a unified framework that encompasses previous tilting theories in various categorical contexts.

Contribution

It defines tilting and cotilting subcategories in extriangulated categories, establishing a Bazzoni characterization and an Auslander-Reiten correspondence with certain subcategories.

Findings

Unifies tilting theory across module and abelian categories.

Extends results to triangulated categories with proper classes of triangles.

Provides a framework connecting tilting subcategories with resolving and coresolving subcategories.

Abstract

Extriangulated category was introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (or cotilting) subcategories in an extriangulated category is defined in this paper. We give a Bazzoni characterization of tilting (or cotilting) subcategories and obtain an Auslander-Reiten correspondence between tilting (cotilting) subcategories and coresolving covariantly (resolving contravariantly, resp.) finite subcatgories which are closed under direct summands and satisfies some cogenerating (generating, resp.) conditons. Applications of the results are given: we show that tilting (cotilting) subcategories defined here unify many previous works about tilting theory in module categories of Artin algebras and abelian categories admitting a cotorsion triples; we also show that the results work for triangulated categories with a proper class of triangles introduced by Beligiannis.