Wide subcategories and lattices of torsion classes

In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category A from the point of view of lattice theory. Motivated by τ-tilting reduction of Jasso, we mainly focus on intervals [U, T] in the lattice torsA of torsion classes in A such that W: = U^⊥∩ T is a wide subcategory of A; we call these intervals wide intervals. We prove that a wide interval [U, T] is isomorphic to the lattice torsW of torsion classes in the abelian category 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–Šťovíček.