Grothendieck monoids of extriangulated categories

TL;DR

This paper explores the Grothendieck monoid of extriangulated categories, providing new classifications and isomorphism results, with applications to derived categories and torsionfree classes.

Contribution

It introduces classifications of subcategories via the Grothendieck monoid and establishes isomorphisms for localizations, extending known results to extriangulated categories.

Findings

Classified Serre and dense 2-out-of-3 subcategories using the Grothendieck monoid

Proved isomorphism of Grothendieck monoids under localization in certain cases

Computed Grothendieck monoids for intermediate subcategories and related torsionfree classes

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.