Stratifying systems and Jordan-Hölder extriangulated categories

Thomas Brüstle, Souheila Hassoun, Amit Shah, Aran Tattar

Published 2022-08-16Version 1

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 $\Phi$ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory $\mathcal{F}(\Phi)$ of objects admitting a composition series-like filtration with factors in $\Phi$ has the Jordan-H\"{o}lder 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-H\"{o}lder 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 $\Phi$ in an extriangulated category is part of a projective one $(\Phi, Q)$. We prove that $\mathcal{F}(\Phi)$ is a Jordan-H\"{o}lder extriangulated category when $(\Phi,Q)$ satisfies some extra conditions.