Quotient categories of extriangulated categories

Abstract

Let $\mathcal{A}$ be a subcategory of an extriangulated category $\mathcal{C}$ and let $\mathcal{X}$ be a strongly functorially finite subcategory of $\mathcal{A}.$ Then the subfactor category $\mathcal{A}/\mathcal{X}$ admits a pre-triangulated structure in the sense of Beligiannis-Reiten where $\mathcal{A}$ has the following properties: $(\mathbf{C}\mathbf{2})$ For any $\mathbb{E}$-triangle $A\stackrel{f}{\to }B\to C\mathrm{}--\to$ in $\mathcal{C}$ where $f:A\to B\in \mathcal{A}$ is $\mathcal{X}$-monic, the third term C is also in $\mathcal{A};$ $(\mathbf{C}\mathbf{2}\prime )$ For any $\mathbb{E}$-triangle $A\to B\stackrel{g}{\to }C\mathrm{}--\to$ in $\mathcal{C}$ where $g:B\to C\in \mathcal{A}$ is $\mathcal{X}$-epic, the first term A is also in $\mathcal{A}.$ Moreover, if the above pre-triangulated category $\mathcal{A}/\mathcal{X}$ with $\mathbb{E}(\mathcal{X},\mathcal{X})=0$ becomes a triangulated category, then $(\mathcal{A},\mathcal{A})$ forms an $\mathcal{X}$-mutation pair and $\mathcal{A}$ is closed under extensions.

Keywords:

• Extriangulated category
• mutation pair
• quotient category
• strongly functorially finite subcategory

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 18E10 18E99 18G05

Acknowledgement

We sincerely thank the referee for his or her helpful comments and valuable suggestions improving this paper.

Additional information

Funding

Supported by the National Natural Science Foundation of China (Grant No. 11771212, No. 11801004) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.