## 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)].

### MSC:

 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

### Keywords:

Grothendieck group; $$K$$-group; $$n$$-sequence

### Citations:

Zbl 1291.18015; Zbl 1276.19003
Full Text:

### References:

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