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

Published in Adv. Math., December 2021. Links: journal, arXiv:2005.01626

Citation: H. Enomoto, Monobrick, a uniform approach to torsion-free classes and wide subcategories, Adv. Math. 393 (2021), Paper No. 108113, 41 pp.

## Comment

It is well known that semibricks are in bijection with wide (=extension-closed exact abelian) subcategory in a length abelian category, by considering simple objects in wide subcats. I continue studying simples in torsion-free classes, and find that similar classification can be achieved by weakening semibrick to monobrick, a set of bricks in which every non-zero morphism is an injection. By this, I can prove that torsion-free classes are in bijection with *cofinally closed monobrick, a monobrick satisfying some poset theoretical condition, without any assumption on functorially finiteness, without using tau-tilting theory. This enables us to consider wide subcats and torsion-free classes simultaneously, and several results like a bijection between wide and torf, a finiteness of torf and bricks etc., can be proved poset theoretically or combinatorially.