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 $\mathcal {A}$ from point view lattice theory. Motivated by ?-tilting reduction Jasso, mainly focus on intervals $[\mathcal {U},\mathcal {T}]$ in $\operatorname {\mathsf {tors}} \mathcal such that {W}:=\mathcal {U}^{\perp } \cap {T}$ is a subcategory ; call these intervals. We prove interval isomorphic to {W}$ . also characterize two ways: First, purely theoretic terms based brick labeling established Demonet–Iyama–Reading–Reiten–Thomas; second, Ingalls–Thomas correspondences subcategories, which were further developed Marks–Š?oví?ek.