Semibricks in extriangulated categories

TL;DR

This paper establishes new correspondences between simple semibricks, wide subcategories, and cotorsion pairs in extriangulated categories, generalizing previous results from module and exact categories, and applies these to triangulated categories.

Contribution

It introduces a novel bijection between simple semibricks and wide subcategories, extending classical results to the setting of extriangulated categories.

Findings

One-to-one correspondence between simple semibricks and wide subcategories.

Correspondence between cotorsion pairs and subsets of semibricks.

Application to simple minded systems in triangulated categories.

Abstract

Let $\mathcal{X}$ be a semibrick in an extriangulated category $\mathscr{C}$. Let $\mathcal{T}$ be the filtration subcategory generated by $\mathcal{X}$. We give a one-to-one correspondence between simple semibricks and length wide subcategories in $\mathscr{C}$. This generalizes a bijection given by Ringel in module categories, which has been generalized by Enomoto to exact categories. Moreover, we also give a one-to-one correspondence between cotorsion pairs in $\mathcal{T}$ and certain subsets of $\mathcal{X}$. Applying to the simple minded systems of an triangulated category, we recover a result given by Dugas.