Extriangulated length categories: torsion classes and $τ$-tilting theory
Li Wang, Jiaqun Wei, Haicheng Zhang, Panyue Zhou
TL;DR
The paper develops extriangulated length categories $(\mathcal{A},\Theta)$, showing they are precisely those with a simple-minded system and extending classical length-category theory to the extriangulated setting. It then builds a rich lattice-theoretic framework for torsion classes, proving that the torsion lattice is complete, completely semidistributive, and algebraic, with brick-based descriptions of intervals and a brick labeling of the Hasse quiver. A key contribution is the extension of semibrick–length-wide subcategory correspondences and the brick-labeled torsion structure to extriangulated length categories, unifying abelian, exact, and triangulated contexts. Finally, the authors generalize tau-tilting theory by introducing support torsion classes and support tau-tilting subcategories, establishing a bijection between them and extending the Adachi–Iyama–Reiten correspondence to this broad setting, provided there are enough Theta-projectives.
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 ${\rm tor}_Θ(\mathcal{A})$ of torsion classes in an extriangulated length category $(\mathcal{A},Θ)$ from the perspective of lattice theory. It is shown that ${\rm tor}_Θ(\mathcal{A})$ forms a complete lattice, which is further proved to be completely semidistributive and algebraic. Moreover, we describe the arrows in the Hasse quiver of ${\rm tor}_Θ(\mathcal{A})$ using brick labeling. Finally, we introduce the concepts of support torsion classes and support $τ$-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.