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

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 latticetheoretical 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. Contents 1. Introduction 1 2. Preliminaries 4 2.1. The kappa map and the join-irreducible labeling 5 2.2. The kappa map in the lattice of torsion classes 7 3. Computing the posets of torsion hearts 9 3.1. Preliminaries on torsion hearts 9 3.2. Construction 10 3.3. Posets of subcategories as posets of join-irreducibles 11 3.4. The map j-label in terms of the join-irreducible labeling 12 4. The kappa order, the core label order, and wide A 13 4.1. Canonical join representation and the extended kappa map 13 4.2. Canonical join representation and widely generated torsion classes 16 4.3. The kappa order and wide subcategories 19 4.4. The core label order and wide subcategories 20 4.5. Combinatorial consequences 23 References 24

Monobrick, a uniform approach to torsion-free classes and wide subcategories

“…Proof. The proof of this claim has already been established in [En3,Lemma 4.23] under the assumption that A is an abelian length category. However, since the proof therein only depends on the descriptions of T(C) and F(C) given in Lemmas 2.5 and 2.6, respectively, the same proof applies in this general setting.…”

Semidistrim Lattices

“…Remark 11.8. After we released the original preprint of this article, the preprint [22] independently defined the pop-core and row-core label orders for semidistributive lattices. See also [6].…”