From the Lattice of Torsion Classes to the Posets of Wide Subcategories and ICE-closed Subcategories

In this paper, we compute the posets of wide subcategories and ICE-closed subcategories from the lattice of torsion classes in an abelian length category in a purely lattice-theoretical way, by using the kappa map in a completely semidistributive lattice. As for the poset of wide subcategories, we give two more simple constructions via a bijection between wide subcategories and torsion classes with canonical join representations. More precisely, for a completely semidistributive lattice, we give two poset structures on the set of elements with canonical join representations: the kappa order (defined using the extended kappa map of Barnard–Todorov–Zhu), and the core label order (generalizing the shard intersection order for congruence-uniform lattices). Then we show that these posets for the lattice of torsion classes coincide and are isomorphic to the poset of wide subcategories. As a byproduct, we give a simple description of the shard intersection order on a finite Coxeter group using the extended kappa map.