Based on paper:
In this talk, I introduce a monobrick in a length abelian category, a set of bricks such that every non-zero morphism between them are injections. This generalizes semibricks. By taking simple objects and extension-closures, monobricks bijectively correspond to left Schur subcategories of an abelian category, which unify both torsion-free classes and wide subcategories. In particular, this enables us to classify all torsion-free classes in any length abelian category by cofinally closed monobricks. Then I’ll explain that several known results, such as Demonet-Iyama-Jasso and Marks-Stovicek, can be easily deduced by using only poset-theoretical argument of monobricks, without using tau-tilting theory, hence relations between wide subcategories and torsion-free classes become more transparent.