The Grothendieck Group of an $n$-exangulated Category

TL;DR

This paper introduces the Grothendieck group for n-exangulated categories, extending classical results and classifying dense subcategories via subgroups, thereby unifying several categorical frameworks.

Contribution

It defines the Grothendieck group for n-exangulated categories and classifies dense subcategories with n-(co)generators, extending existing theories to a broader categorical context.

Findings

The Grothendieck group of an n-exangulated category shares properties with classical Grothendieck groups.

Dense subcategories with n-(co)generators correspond to subgroups of the Grothendieck group.

The classification unifies results across triangulated, (n+2)-angulated, exact, and extriangulated categories.

Abstract

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, $(n+2)$-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.