arxiv: 2604.23352 · v1 · submitted 2026-04-25 · 🧮 math.RT · math.CT

Recognition: unknown

Chains of model structures arising from cotorsion pairs on extriangulated categories

Dandan Sun, Dongdong Zhang, Haiyan Zhu, Panyue Zhou, Xiaoyan Yang

Pith reviewed 2026-05-08 07:01 UTC · model grok-4.3

classification 🧮 math.RT math.CT

keywords cotorsion pairsextriangulated categoriesHovey triplesmodel structureshomotopy categoriesGorenstein injective dimensionstable categoriestriangulated equivalences

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The pith

From one hereditary Hovey triple on an extriangulated category, further triples are built whose homotopy categories are equivalent, producing a chain of model structures all triangulated equivalent to one stable category.

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

The paper establishes a method to start with a hereditary Hovey triple coming from cotorsion pairs and generate additional hereditary Hovey triples. Under suitable completeness assumptions on the pairs, the associated homotopy categories become triangulated equivalent. This yields an infinite chain of model structures on the category whose homotopy categories all reduce to the same stable category. The construction is applied to the class of objects with finite Gorenstein injective dimension defined by a proper class of E-triangles, recovering classical results on Gorenstein injective modules and supplying new examples inside derived categories via cohomological ghost triangles.

Core claim

Starting with a hereditary Hovey triple, further hereditary Hovey triples are constructed whose homotopy categories are equivalent under suitable completeness assumptions on the cotorsion pairs. As an application, a chain of model structures is obtained whose homotopy categories are all triangulated equivalent to a common stable category, recovering known results for Gorenstein injective modules and yielding new examples in the derived category of a ring when the proper class is given by cohomological ghost triangles.

What carries the argument

hereditary Hovey triple induced by a pair of cotorsion pairs, which supplies a model structure on the extriangulated category whose homotopy category is the stable category

If this is right

The homotopy categories of the successive hereditary Hovey triples are triangulated equivalent.
The resulting model structures form a chain whose homotopy categories coincide with one fixed stable category.
The same stable category arises as the homotopy category for the model structures associated to objects of finite Gorenstein injective dimension.
New model structures appear on the derived category of a ring when the proper class consists of cohomological ghost triangles.

Where Pith is reading between the lines

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

The chain construction may be iterable, allowing arbitrarily long sequences of model structures from a single starting triple.
The equivalence of homotopy categories could extend the comparison of stable categories across different proper classes of triangles.
The method supplies a uniform way to produce multiple Quillen equivalent model structures on the same underlying extriangulated category.

Load-bearing premise

The constructions and equivalences require suitable completeness assumptions on the cotorsion pairs together with mild set-theoretic assumptions.

What would settle it

An explicit cotorsion pair satisfying the hereditary condition but lacking the stated completeness properties for which no chain of equivalent homotopy categories exists would falsify the main claim.

read the original abstract

The main aim of this paper is to study chains of model structures arising from cotorsion pairs in extriangulated categories. Starting with a hereditary Hovey triple, we construct further hereditary Hovey triples whose homotopy categories are equivalent under suitable completeness assumptions, thereby refining results due to El Maaouy and Shao-Wang-Zhang. As an application, we consider objects of finite Gorenstein injective dimension with respect to a proper class of $\mathbb{E}$-triangles. Under mild set-theoretic assumptions, we obtain a chain of model structures whose homotopy categories are all triangulated equivalent to a common stable category. This recovers known results for Gorenstein injective modules and yields new examples in the derived category of a ring when the proper class is given by cohomological ghost triangles.

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.

Desk Editor's Note private letter to a colleague

This paper builds explicit chains of hereditary Hovey triples from one starting triple in extriangulated categories and uses them to produce new model structures on derived categories via ghost triangles.

read the letter

The central contribution is a construction that starts with a hereditary Hovey triple and iteratively produces further ones whose model structures have homotopy categories triangulated equivalent to a single stable category. This refines the results of El Maaouy and Shao-Wang-Zhang by making the chain explicit and by applying it to proper classes of E-triangles, including the case of cohomological ghost triangles. The application recovers the known model structures for Gorenstein injective modules and gives fresh examples on the derived category of a ring under mild set-theoretic assumptions. The paper does this by defining new cotorsion pairs from the original data, checking that the triples stay hereditary, and building the necessary comparison functors. That part is useful for specialists who already handle cotorsion pairs and Hovey triples in extriangulated settings. The stress-test note confirms the steps are internally consistent once the completeness hypotheses are granted, with no evident circularity or axiom failure in the outline. The main limitation is the reliance on those completeness assumptions for the cotorsion pairs; if they do not hold, the chain and the equivalences simply do not appear. The abstract gives no proofs, so the full manuscript must supply the closure verifications and the explicit functor comparisons. No other soft spots stand out. This work is aimed at readers already comfortable with model structures on triangulated and extriangulated categories. A broader algebraist or someone outside homological algebra will not find much to use. It deserves a serious referee because the chain technique and the ghost-triangle examples are concrete, build directly on cited prior results, and can be checked independently in specific cases.

Referee Report

0 major / 3 minor

Summary. The paper studies chains of model structures induced by cotorsion pairs in extriangulated categories. Starting from a hereditary Hovey triple, it constructs further hereditary Hovey triples whose associated model structures have homotopy categories that are triangulated-equivalent under suitable completeness assumptions on the cotorsion pairs. This refines results of El Maaouy and Shao-Wang-Zhang. As an application, for objects of finite Gorenstein injective dimension with respect to a proper class of E-triangles (including cohomological ghost triangles), the paper produces a chain of model structures whose homotopy categories are all triangulated-equivalent to a common stable category, recovering known results for Gorenstein injective modules and yielding new examples in derived categories of rings.

Significance. If the constructions and equivalences hold, the work provides a systematic iterative method for generating chains of model structures with equivalent homotopy categories in the extriangulated setting. It extends prior results on Hovey triples and cotorsion pairs while supplying concrete applications that recover classical Gorenstein homological algebra and produce new triangulated equivalences in derived categories. The explicit comparison functors and closure properties under the stated hypotheses strengthen the toolkit for studying stable categories via model-categorical methods.

minor comments (3)

The introduction should explicitly list the precise completeness conditions (e.g., the closure properties of the cotorsion pairs under the E-triangles) that are invoked in the main theorems, rather than referring only to 'suitable completeness assumptions'.
In the application section on finite Gorenstein injective dimension, clarify whether the proper class of E-triangles is assumed to be closed under extensions or only under the given mild set-theoretic hypotheses; a short remark would prevent ambiguity when recovering the module case.
Notation for the successive cotorsion pairs and the induced Hovey triples (e.g., the indexing of the chain) could be made uniform across statements to ease comparison with the cited works of El Maaouy and Shao-Wang-Zhang.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on chains of model structures arising from cotorsion pairs in extriangulated categories, their recognition of the refinements to results of El Maaouy and Shao-Wang-Zhang, and the applications recovering Gorenstein results while providing new examples in derived categories. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity; constructions are explicit and self-contained

full rationale

The paper starts from a given hereditary Hovey triple and explicitly constructs further ones by defining new cotorsion pairs from the original data, then verifies heredity and homotopy equivalences via functors under stated completeness assumptions. No equations reduce outputs to inputs by definition, no parameters are fitted and renamed as predictions, and cited prior results (El Maaouy, Shao-Wang-Zhang) are external refinements rather than load-bearing self-citations. The chain of model structures and triangulated equivalences to a stable category follow from the iterative definitions and closure properties supplied in the text, making the derivation independent of its own conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard axioms of extriangulated categories and the existence of hereditary Hovey triples; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption Extriangulated categories admit cotorsion pairs and hereditary Hovey triples with the usual orthogonality and completeness properties.
Invoked as the starting point for constructing further triples; standard background from the theory of extriangulated categories.
domain assumption Completeness assumptions on the cotorsion pairs guarantee the existence of the required model structures and equivalences of homotopy categories.
Explicitly stated as necessary for the chain construction and the triangulated equivalences.

pith-pipeline@v0.9.0 · 5439 in / 1507 out tokens · 92474 ms · 2026-05-08T07:01:07.326911+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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