We define the Grothendieck group of an n -exangulated category. For n odd, we show that this group shares many properties with the Grothendieck group of an exact or a triangulated category. In particular, we classify dense complete subcategories of an n -exangulated category with an n -(co)generator in terms of subgroups of the Grothendieck group. This unifies and extends results of Thomason, Bergh–Thaule, Matsui and Zhu–Zhuang for triangulated, $$(n+2)$$ ( n + 2 ) -angulated, exact and extriangulated categories, respectively. We also introduce the notion of an n -exangulated subcategory and prove that the subcategories in our classification theorem carry this structure.

中文翻译：

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n-exangulated 范畴的格洛腾迪克群

我们定义了一个 n-exangulated 范畴的 Grothendieck 群。对于 n 奇数，我们证明该群与精确或三角范畴的 Grothendieck 群共享许多性质。特别地，我们根据 Grothendieck 群的子群对具有 n -(co) 生成器的 n -exangulated 类别的密集完全子类别进行分类。这统一并扩展了 Thomason、Bergh-Thaule、Matsui 和 Zhu-Zhuang 分别用于三角化、$$(n+2)$$ ( n + 2 ) 角化、精确和外三角化类别的结果。我们还引入了一个 n-exangulated 子类别的概念，并证明了我们分类定理中的子类别具有这种结构。