  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.
