For a triangulated category $${\mathcal {T}}$$ T , if $${\mathcal {C}}$$ C is a cluster-tilting subcategory of $${\mathcal {T}}$$ T , then the factor category $${\mathcal {T}}{/}{\mathcal {C}}$$ T / C is an abelian category. Under certain conditions, the converse also holds. This is a very important result of cluster-tilting theory, due to Koenig–Zhu and Beligiannis. Now let $${\mathcal {B}}$$ B be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory $${\mathcal {C}}$$ C of $${\mathcal {B}}$$ B , which is a generalization of cluster tilting subcategory. We show that $${\mathcal {C}}$$ C is cluster tilting if and only if the factor category $${\mathcal {B}}{/}{\mathcal {C}}$$ B / C is abelian. Our result generalizes the related results on a triangulated category and is new for an exact category case.

中文翻译：

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由簇倾斜子类别产生的阿贝尔类别

对于三角化类别 $${\mathcal {T}}$$ T ，如果 $${\mathcal {C}}$$ C 是 $${\mathcal {T}}$$ T 的集群倾斜子类别，那么因子范畴 $${\mathcal {T}}{/}{\mathcal {C}}$$ T / C 是一个阿贝尔范畴。在一定条件下，反之亦然。由于 Koenig-Zhu 和 Beligiannis，这是集群倾斜理论的一个非常重要的结果。现在让 $${\mathcal {B}}$$ B 是一个合适的外三角化类别，它是三角化类别和精确类别的同时推广。我们引入了 $${\mathcal {B}}$$ B 的预聚类倾斜子类别 $${\mathcal {C}}$$ C 的概念，它是聚类倾斜子类别的推广。我们证明 $${\mathcal {C}}$$ C 是集群倾斜当且仅当因子类别 $${\mathcal {B}}{/}{\mathcal {C}}$$ B / C 是阿贝尔。