Auslander--Reiten theory in extriangulated categories

TL;DR

This paper extends Auslander--Reiten theory to extriangulated categories, unifying existing theories and establishing conditions for almost split extensions, with implications for stable categories and quiver representations.

Contribution

It develops Auslander--Reiten theory within extriangulated categories, unifying prior theories and exploring conditions for almost split extensions and their applications.

Findings

Existence of almost split extensions is equivalent to Auslander--Reiten--Serre duality under certain conditions.

Stable categories of extriangulated categories are τ-categories with various structural consequences.

Any locally finite symmetrizable τ-quiver can be realized as an Auslander--Reiten quiver of an extriangulated category.

Abstract

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander--Reiten theory for extriangulated categories. This unifies Auslander--Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander--Reiten--Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category $\underline{\mathscr{C}}$ of an extriangulated category $\mathscr{C}$ is a $\tau$-category if $\mathscr{C}$ has enough projectives, almost split extensions and source morphisms. This gives various consequences on $\underline{\mathscr{C}}$, including Igusa--Todorov's Radical Layers Theorem, Auslander--Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of $\underline{\mathscr{C}}$ via the complete mesh category of the Auslander--Reiten species of $\underline{\mathscr{C}}$. Finally we prove that any locally finite symmetrizable $\tau$-quiver (=valued translation quiver) is an Auslander--Reiten quiver of some extriangulated category with sink morphisms and source morphisms.