Grothendieck monoids of extriangulated categories

Abstract

We study the Grothendieck monoid (a monoid version of the Grothendieck group) of an extriangulated category, and give some results which are new even for abelian categories. First, we classify Serre subcategories and dense 2-out-of-3 subcategories using the Grothendieck monoid. Second, in good situations, we show that the Grothendieck monoid of the localization of an extriangulated category is isomorphic to the natural quotient monoid of the original Grothendieck monoid. This includes the cases of the Serre quotient of an abelian category and the Verdier quotient of a triangulated category. As a concrete example, we introduce an intermediate subcategory of the derived category of an abelian category, which lies between the abelian category and its one shift. We show that intermediate subcategories bijectively correspond to torsionfree classes in the abelian category, and then compute the Grothendieck monoid of an intermediate subcategory. Contents 24 5. Intermediate subcategories of the derived category 25 5.1. Classification of intermediate subcategories 25 5.2. Grothendieck monoid of an intermediate subcategory 27 5.3. Serre subcategories of an intermediate subcategory 29 6. Questions 34 6.1. Invertible elements in the Grothendieck monoid 34 6.2. Quasi-split extriangulated categories and c-closed subcategories 35 Appendix A. Basics on commutative monoids 37 References 41