Grothendieck group for sequences. (English) Zbl 1481.18016

Summary: For any category with a distinguished collection of sequences, such as \(n\)-exangulated category, category of \(N\)-complexes and category of precomplexes, we consider its Grothendieck group and similar results of P. A. Bergh and M. Thaule for \(n\)-angulated categories [J. Pure Appl. Algebra 218, No. 2, 354–366 (2014; Zbl 1291.18015)] are proven. A classification result of dense complete subcategories is given and we give a formal definition of \(K\)-groups for these categories following D. R. Grayson’s algebraic approach of \(K\)-theory for exact categories [J. Am. Math. Soc. 25, No. 4, 1149–1167 (2012; Zbl 1276.19003)].


18F30 Grothendieck groups (category-theoretic aspects)
18G80 Derived categories, triangulated categories
18E10 Abelian categories, Grothendieck categories
19A99 Grothendieck groups and \(K_0\)
19D99 Higher algebraic \(K\)-theory
Full Text: DOI


[1] P. A. Bergh and M. Thaule, The Grothendieck group of an n-angulated category, J. Pure Appl. Algebra 218 (2014), no. 2, 354-366. https://doi.org/10.1016/j.jpaa.2013.06. 007 · Zbl 1291.18015
[2] B. I. Dundas, T. G. Goodwillie, and R. McCarthy, The local structure of algebraic K-theory, Algebra and Applications, 18, Springer-Verlag London, Ltd., London, 2013.
[3] C. Geiss, B. Keller, and S. Oppermann, n-angulated categories, J. Reine Angew. Math. 675 (2013), 101-120. https://doi.org/10.1515/crelle.2011.177 · Zbl 1271.18013
[4] D. R. Grayson, Algebraic K-theory via binary complexes, J. Amer. Math. Soc. 25 (2012), no. 4, 1149-1167. https://doi.org/10.1090/S0894-0347-2012-00743-7 · Zbl 1276.19003
[5] T. Harris, Algebraic proofs of some fundamental theorems in algebraic K-theory, Homol-ogy Homotopy Appl. 17 (2015), no. 1, 267-280. https://doi.org/10.4310/HHA.2015. v17.n1.a13 · Zbl 1375.19007
[6] J. Haugland, The Grothendieck group of an n-exangulated category, Appl. Categ. Struc-tures 29 (2021), no. 3, 431-446. https://doi.org/10.1007/s10485-020-09622-w · Zbl 1467.18017
[7] M. Herschend, Y. Liu, and H. Nakaoka, n-exangulated categories (I): Definitions and fundamental properties, J. Algebra 570 (2021), 531-586. https://doi.org/10.1016/j. jalgebra.2020.11.017 · Zbl 07290700
[8] G. Jasso, n-Abelian and n-exact categories, Math. Z. 283 (2016), no. 3-4, 703-759. https://doi.org/10.1007/s00209-016-1619-8 · Zbl 1356.18005
[9] H. Matsui, Classifying Dense Resolving and Coresolving Subcategories of Exact Cate-gories Via Grothendieck Groups, Algebr. Represent. Theory 21 (2018), no. 3, 551-563. https://doi.org/10.1007/s10468-017-9726-8 · Zbl 1400.18016
[10] H. Nakaoka and Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom. Différ. Catég. 60 (2019), no. 2, 117-193. · Zbl 1451.18021
[11] R. W. Thomason, The classification of triangulated subcategories, Compos. Math. 105 (1997), no. 1, 1-27. https://doi.org/10.1023/A:1017932514274 · Zbl 0873.18003
[12] Xuan Yu Public Course Education Department Shenzhen Institute of Information Technology Shenzhen 518172, P. R. China Email address: xuanyumath@outlook.com
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