Semibricks in extriangulated categories

Abstract

Let $\mathcal{X}$ be a semibrick in an extriangulated category $\mathcal{C}.$ 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 $\mathit{C}.$ 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 a triangulated category, we recover a result given by Dugas.

Keywords:

• Cotorsion pairs
• extriangulated categories
• semibricks
• wide subcategories

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 18E05
• 18E30

Acknowledgment

The authors are grateful to the anonymous referees for their helpful comments.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (No. 11801273, No. 11771212) and Natural Science Foundation of Jiangsu Province of China (No. BK20180722).