# Wide subcategories and lattices of torsion classes

## TL;DR

This paper explores the structure of torsion classes and wide subcategories in abelian length categories, establishing lattice isomorphisms and characterizations that deepen understanding of their relationships.

## Contribution

It introduces the concept of wide intervals in the lattice of torsion classes and proves their isomorphism to torsion classes in associated wide subcategories, with new lattice-theoretic characterizations.

## Key findings

- Wide intervals are isomorphic to torsion classes in wide subcategories.
- Characterization of wide intervals via lattice theory and existing correspondences.
- Provides a new perspective on the structure of torsion classes in abelian categories.

## Abstract

In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category $\mathcal{A}$ from the point of view of lattice theory. Motivated by $\tau$-tilting reduction of Jasso, we mainly focus on intervals $[\mathcal{U},\mathcal{T}]$ in the lattice $\operatorname{\mathsf{tors}} \mathcal{A}$ of torsion classes in $\mathcal{A}$ such that $\mathcal{W}:=\mathcal{U}^\perp \cap \mathcal{T}$ is a wide subcategory of $\mathcal{A}$; we call these intervals wide intervals. We prove that a wide interval $[\mathcal{U},\mathcal{T}]$ is isomorphic to the lattice $\operatorname{\mathsf{tors}} \mathcal{W}$ of torsion classes in the abelian category $\mathcal{W}$. We also characterize wide intervals in two ways: First, in purely lattice theoretic terms based on the brick labeling established by Demonet--Iyama--Reading--Reiten--Thomas; and second, in terms of the Ingalls--Thomas correspondences between torsion classes and wide subcategories, which were further developed by Marks--\v{S}\v{t}ov\'{i}\v{c}ek.