Let $(C,E,s)$ be an extriangulated category. We show that certain quotient categories of extriangulated categories are equivalent to module categories by some restriction of functor $E$, and in some cases, they are abelian. This result can be regarded as a simultaneous generalization of Koenig-Zhu and Demonet-Liu. In addition, we introduce the notion of maximal rigid subcategories in extriangulated categories. Cluster tilting subcategories are obviously strongly functorially finite maximal rigid subcategories, we prove that the converse is true if the 2-Calabi-Yau extriangulated categories admit a cluster tilting subcategories, which generalizes a result of Buan-Iyama-Reiten-Scott and Zhou-Zhu.
Keywords: Extriangulated categories; Cluster-tilting subcategories; Rigid subcategories; Maximal rigid subcategories; Quotient categories
2010 MSC: 18E30; 18E10
Theory and Applications of Categories, Vol. 34, 2019, No. 8, pp 221-242.