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arxiv: 2208.07808 · v5 · submitted 2022-08-16 · 🧮 math.RT · math.CT

Stratifying systems and Jordan-H\"{o}lder extriangulated categories

Thomas Br\"ustle , Souheila Hassoun , Amit Shah , Aran Tattar This is my paper

Pith reviewed 2026-05-24 11:41 UTC · model grok-4.3

classification 🧮 math.RT math.CT
keywords stratifying systemsextriangulated categoriesJordan-Hölder propertycomposition seriesGrothendieck monoidprojective stratifying systemsfiltration subcategoriesleft exactness condition
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The pith

Every stratifying system in an extriangulated category extends to a minimal projective pair (Φ, Q) such that F(Φ) is a length Jordan-Hölder category when left exactness holds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends stratifying systems, previously studied in module and triangulated categories, to the setting of extriangulated categories. It first introduces subobjects, simple objects, and composition series in extriangulated categories, then uses these to define Jordan-Hölder extriangulated categories and characterize them via the associated Grothendieck monoid and group. The central result establishes that any stratifying system Φ can be completed to a minimal projective stratifying system (Φ, Q), and that the subcategory F(Φ) of objects admitting filtrations with factors in Φ becomes a length category satisfying the Jordan-Hölder property precisely when (Φ, Q) satisfies a left exactness condition. Several examples are constructed and a question of Enomoto-Saito is answered negatively.

Core claim

In an extriangulated category, every stratifying system Φ is contained in a minimal projective stratifying system (Φ, Q). When this pair satisfies a left exactness condition, the subcategory F(Φ) of objects that admit a filtration with successive factors in Φ is a length Jordan-Hölder extriangulated category, meaning every object in F(Φ) has a composition series and any two such series have the same length with factors isomorphic up to permutation and repetition.

What carries the argument

The minimal projective stratifying system (Φ, Q) together with the left exactness condition on how the objects in Q interact with the extension bifunctor, which together ensure that filtrations in F(Φ) behave well enough to inherit the Jordan-Hölder property.

If this is right

  • F(Φ) is always a length Jordan-Hölder extriangulated category whenever the left exactness condition holds.
  • Jordan-Hölder extriangulated categories admit a characterization in terms of their Grothendieck monoid and Grothendieck group.
  • Every stratifying system extends to a minimal projective one.
  • Concrete examples of stratifying systems and Jordan-Hölder extriangulated categories exist, including cases that negatively resolve a question of Enomoto-Saito.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The left exactness condition may be necessary rather than merely sufficient; one could search for a counterexample pair (Φ, Q) that is not left exact yet still produces a Jordan-Hölder F(Φ).
  • The construction supplies a method for producing length categories inside arbitrary extriangulated categories, which could be applied to study filtrations in other homological settings such as stable categories or derived categories.
  • The negative answer to the Enomoto-Saito question indicates that additional structural hypotheses beyond those previously considered are needed to guarantee the Jordan-Hölder property in general.

Load-bearing premise

The left exactness condition on the minimal projective pair (Φ, Q) must hold for F(Φ) to be a Jordan-Hölder category.

What would settle it

An explicit extriangulated category together with a stratifying system Φ and projective objects Q forming a left-exact pair (Φ, Q) in which some object of F(Φ) admits two composition series of unequal length or with non-isomorphic factors up to permutation.

Figures

Figures reproduced from arXiv: 2208.07808 by Amit Shah, Aran Tattar, Souheila Hassoun, Thomas Br\"ustle.

Figure 5.1
Figure 5.1. Figure 5.1: Segment of the Auslander-Reiten quiver of C. Lastly, we let A denote some full, extension-closed subcategory of C containing the indecompos￾able objects depicted in [PITH_FULL_IMAGE:figures/full_fig_p027_5_1.png] view at source ↗
read the original 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 $\Phi$ 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 for an extriangulated category, in order to define a Jordan-H\"{o}lder extriangulated category. Moreover, we characterise Jordan-H\"{o}lder, length, weakly idempotent complete extriangulated 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 minimal projective one $(\Phi,Q)$. We prove that $\mathcal{F}(\Phi)$ is a length, Jordan-H\"{o}lder extriangulated category when $(\Phi,Q)$ satisfies a left exactness condition. We give several examples and answer a recent question of Enomoto--Saito in the negative.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript extends stratifying systems to extriangulated categories. It defines subobjects, simple objects and composition series to introduce Jordan-Hölder extriangulated categories, characterizes length and Jordan-Hölder properties via the associated Grothendieck monoid and group, constructs projective stratifying systems, proves that every stratifying system Φ belongs to a minimal projective completion (Φ, Q), and shows that F(Φ) is a length Jordan-Hölder extriangulated category precisely when the pair (Φ, Q) satisfies a left exactness condition. Examples are supplied and a question of Enomoto–Saito receives a negative answer.

Significance. The work supplies a uniform categorical framework that recovers and generalizes earlier results for modules, triangulated categories and exact categories. The Grothendieck-monoid characterization and the explicit negative answer to an open question are concrete contributions that may facilitate further study of filtrations and stratifications in extriangulated settings.

minor comments (2)
  1. The precise statement of the left exactness condition on (Φ, Q) (mentioned in the abstract) should be recalled explicitly when it is first used in the proof that F(Φ) is Jordan-Hölder, to make the dependence on this hypothesis transparent.
  2. A short comparison paragraph relating the new notions of subobject and composition series to the corresponding notions already present in the triangulated and exact cases would improve readability for readers coming from those settings.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments appear in the report, so we have no points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained from category axioms

full rationale

The paper defines new notions (subobjects, simple objects, composition series, Jordan-Hölder extriangulated categories) directly from the extriangulated structure, extension bifunctor, and associated Grothendieck monoid/group. The two main results—every stratifying system extends to a minimal projective pair (Φ,Q), and F(Φ) inherits the Jordan-Hölder property under an explicit left exactness hypothesis—are proved from these definitions and the ambient category axioms. No equations reduce a claimed prediction to a fitted input by construction, no uniqueness theorem is imported from the authors' prior work to force choices, and no ansatz is smuggled via self-citation. Prior literature on stratifying systems supplies context but is not load-bearing for the extriangulated case. The derivation chain therefore remains independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters are introduced or fitted. The work rests on the standard definition of extriangulated categories and the new definitional extensions; no new physical or algebraic entities are postulated.

axioms (1)
  • domain assumption An extriangulated category consists of an additive category equipped with a bifunctor E and a class of conflations satisfying the axioms of extriangulated categories.
    This is the background structure taken from prior literature on which all new definitions and proofs are built.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Weak Waldhausen categories and a localization theorem

    math.KT 2024-06 unverdicted novelty 7.0

    Weak Waldhausen categories are introduced to support one-sided extriangulated localization theorems, yielding right exact K0 sequences, with new proofs of the Enomoto-Saito theorem and a generalization of Sarazola's c...

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