Stratifying systems and Jordan-Hölder extriangulated categories

Abstract

Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system Φ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory F(Φ) of objects admitting a composition series-like filtration with factors in Φ has the Jordan-Hölder property on these filtrations.This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series relative to the extriangulated structure in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise these categories in terms of the associated Grothendieck monoid and Grothendieck group.Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system Φ in an extriangulated category is part of a projective one (Φ, Q). We prove that F(Φ) is a Jordan-Hölder extriangulated category when (Φ, Q) satisfies some extra conditions.