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The Grothendieck Group of an n-exangulated Category

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.

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Acknowledgements

The author would like to thank her supervisor Petter Andreas Bergh for helpful discussions and comments. She would also thank Louis-Philippe Thibault for careful reading and helpful suggestions on a previous version of this paper.

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Correspondence to Johanne Haugland.

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Communicated by Bernhard Keller.

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Haugland, J. The Grothendieck Group of an n-exangulated Category. Appl Categor Struct 29, 431–446 (2021). https://doi.org/10.1007/s10485-020-09622-w

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Keywords

  • Grothendieck group
  • n-exangulated category
  • \((n+2)\)-angulated category
  • n-exact category
  • n-exangulated subcategory
  • Extriangulated subcategory

Mathematics Subject Classification

  • 18E10
  • 18E30
  • 18F30