From the lattice of torsion classes to the posets of wide subcategories and ICE-closed subcategories

TL;DR

This paper develops lattice-theoretical methods to derive the posets of wide and ICE-closed subcategories from torsion classes in abelian categories, introducing new constructions and showing their equivalence.

Contribution

It introduces new lattice-based constructions for posets of wide and ICE-closed subcategories, establishing their isomorphism with torsion class posets in semidistributive lattices.

Findings

Posets of wide subcategories are isomorphic to torsion class posets.

Two new poset structures (kappa order and core label order) are introduced and shown to coincide.

A simple description of shard intersection order on finite Coxeter groups is provided.

Abstract

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.