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Quasi-abelian quotients in extriangulated categories

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Pith's one-line read The quotient of an extriangulated category by a hereditary-type subcategory is quasi-abelian.

desk verdict The paper defines hereditary-type subcategories in extriangulated categories and shows their quotients are quasi-abelian via explicit kernel/cokernel constructions that verify the stability axioms directly. read the letter →

arxiv 2605.26469 v1 pith:GRWPSK6G submitted 2026-05-26 math.RT math.CT

Yu Liu , Yu-Zhe Liu , Panyue Zhou This is my paper · ORCID
classification math.RTmath.CT
keywords extriangulatedcategoriesquasi-abelianhereditary-typesubcategoriesclustertiltingquotientabelianrepresentationtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An extriangulated category with a hereditary-type subcategory W yields a quotient category E/W that is additive and equipped with kernels and cokernels. In this quotient the kernels remain stable under pushouts and the cokernels remain stable under pullbacks, which is the definition of a quasi-abelian category. The same construction produces an abelian category precisely when W is cluster tilting inside a suitable relative extriangulated structure on E. This supplies a uniform way to obtain both recovered abelian hearts and new quasi-abelian examples from extriangulated data.

What carries the argument

A hereditary-type subcategory W, which forces the kernels and cokernels of the quotient E/W to satisfy the required stability under pushouts and pullbacks.

What would settle it

An extriangulated category containing a hereditary-type subcategory W for which some kernel in E/W fails to be stable under a pushout would falsify the claim.

Watch

Extended reading notes

Core claim

Let (E, E, s) be an extriangulated category and let W be a hereditary-type subcategory of E. The quotient E/W is then a quasi-abelian category. Moreover, E/W is abelian if and only if W is a cluster tilting subcategory in a suitable relative extriangulated structure.

Load-bearing premise

The subcategory W satisfies the hereditary-type conditions that guarantee stability of kernels and cokernels after quotienting.

Editorial extensions

If this is right

  • E/W always carries kernels stable under pushouts and cokernels stable under pullbacks.
  • The quotient becomes abelian exactly when W is cluster tilting in the indicated relative structure.
  • The construction recovers known abelian hearts and produces quasi-abelian quotients outside classical triangulated or exact settings.
  • Several concrete examples confirm that the quotient operation works in both recovered and previously unseen cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same quotient construction may be tested on concrete extriangulated categories arising from cluster categories or derived categories of hereditary algebras.
  • Links to tilting theory suggest that further examples could be obtained by varying the relative extriangulated structure on W.
  • If the hereditary-type condition can be verified algorithmically in finite-type cases, the method would give an explicit source of new quasi-abelian categories.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper introduces the notion of a hereditary-type subcategory W in an extriangulated category (E, E, s). It proves that the quotient E/W is quasi-abelian (additive with kernels and cokernels stable under pushouts and pullbacks respectively). It further shows that E/W is abelian if and only if W is cluster-tilting in a suitable relative extriangulated structure, and supplies examples recovering classical abelian hearts as well as new quotients.

Significance. If the results hold, the work supplies an explicit construction of kernels and cokernels in the quotient together with direct verification of the stability axioms from the hereditary-type conditions; this yields a systematic method for producing quasi-abelian and abelian categories from extriangulated data, extending the hereditary-algebra case and linking to cluster-tilting theory. The parameter-free character of the stability statements and the recovery of known examples are particular strengths.

minor comments (3)
  1. [§2] §2: the definition of hereditary-type subcategory would be clearer if it included an explicit statement of the two or three axioms that are used in the subsequent pushout/pullback verifications.
  2. [main theorem] The statement of the main theorem (presumably Theorem 3.1 or 3.2) should record the precise relative extriangulated structure in which the cluster-tilting condition is formulated.
  3. [examples] In the examples, the verification that the induced kernels and cokernels in E/W coincide with the quotient constructions could be expanded by one sentence each.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in constructing quasi-abelian and abelian categories from extriangulated data, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper introduces a new definition of hereditary-type subcategory W and derives the quasi-abelian property of the quotient E/W directly from the extriangulated axioms plus the new definition's stability conditions. Kernels and cokernels in the quotient are constructed explicitly, and pushout/pullback stability is verified from those axioms without reducing to any fitted parameter, self-referential equation, or prior self-citation that bears the load. The abelian iff statement is likewise a direct equivalence to a cluster-tilting condition in a relative structure, illustrated by examples that recover known cases. No step equates a claimed prediction or uniqueness result to its own input by construction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The paper's main results rely on the newly defined hereditary-type subcategory and the axioms of extriangulated categories from prior work. No numerical parameters are fitted.

assumptions (2)
  • standard math Standard axioms of category theory and additive categories
    The paper works within the framework of extriangulated categories which rely on these.
  • domain assumption Properties of extriangulated categories as defined in prior literature
    The structure (E, E, s) is assumed to satisfy the extriangulated axioms.
invented entities (1)
  • hereditary-type subcategory
    purpose: To ensure the quotient is quasi-abelian
    This is a new definition introduced in the paper to make the main theorem hold.

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Cite this review

Pith. "Pith review of Quasi-abelian quotients in extriangulated categories." pith.science (2026). https://pith.science/paper/GRWPSK6G

@misc{pith2026260526469,
  author       = {Pith},
  title        = {Pith review of: Quasi-abelian quotients in extriangulated categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRWPSK6G}},
  note         = {Machine review of arXiv:2605.26469}
}
abstract

Let $(\mathcal{E}, \mathbb{E}, \mathfrak{s})$ be an extriangulated category. Motivated by the theory of hereditary algebras, we introduce the notion of a hereditary-type subcategory $\mathcal{W}\subseteq \mathcal{E}$. We prove that the quotient $\mathcal{E}/\mathcal{W}$ is a quasi-abelian category, that is, an additive category with kernels and cokernels in which kernels are stable under pushouts and cokernels are stable under pullbacks. Moreover, we show that $\mathcal{E}/\mathcal{W}$ is abelian if and only if $\mathcal{W}$ is a cluster tilting subcategory in a suitable relative extriangulated structure. Several examples are provided to illustrate the main results, showing that our approach both recovers known abelian hearts and yields new abelian or quasi-abelian quotients beyond classical settings.

Figures

Figures reproduced from arXiv: 2605.26469 by the authors.

Figure 1
Figure 1. A connected component of Auslander–Reiten quiver of A = kQ Auslander–Reiten quiver of A. Take E = modA and W = ⟨W ∈ ind(modA) | W ̸∼= [(2t) +, 1,(2t) −] (t ∈ N +)⟩ (see the shadow in [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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