Extriangulated length categories: torsion classes and $\tau$-tilting theory

TL;DR

This paper develops the theory of extriangulated length categories, characterizes when they are length categories, and explores their torsion classes and $ au$-tilting theory, generalizing existing bijections.

Contribution

It introduces extriangulated length categories, characterizes them via simple-minded systems, and extends $ au$-tilting theory to this setting with new bijections.

Findings

The torsion classes form a complete, semidistributive, algebraic lattice.

Hasse quiver arrows are described using brick labeling.

A bijection between support torsion classes and support $ au$-tilting subcategories is established.

Abstract

This paper introduces the notion of extriangulated length categories, whose prototypical examples include abelian length categories and bounded derived categories of finite dimensional algebras with finite global dimension. We prove that an extriangulated category $\mathcal{A}$ is a length category if and only if $\mathcal{A}$ admits a simple-minded system. Subsequently, we study the partially ordered set ${\rm tor}_{\Theta}(\mathcal{A})$ of torsion classes in an extriangulated length category $(\mathcal{A},\Theta)$ from the perspective of lattice theory. It is shown that ${\rm tor}_{\Theta}(\mathcal{A})$ forms a complete lattice, which is further proved to be completely semidistributive and algebraic. Moreover, we describe the arrows in the Hasse quiver of ${\rm tor}_{\Theta}(\mathcal{A})$ using brick labeling. Finally, we introduce the concepts of support torsion classes and support $\tau$-tilting subcategories in extriangulated length categories and establish a bijection between these two notions, thereby generalizing the Adachi-Iyama-Reiten bijection for functorially finite torsion classes.