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arxiv: 2503.01188 · v4 · submitted 2025-03-03 · 🧮 math.RT · math.CT· math.RA

Quillen equivalence for chain homotopy categories induced by balanced pairs

Jiangsheng Hu , Wei Ren , Xiaoyan Yang , Hanyang You This is my paper

Pith reviewed 2026-05-23 02:10 UTC · model grok-4.3

classification 🧮 math.RT math.CTmath.RA
keywords balanced pairchain homotopy categoryQuillen equivalencemodel categorytriangulated equivalencecotorsion tripleGorenstein projectivepure injective
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The pith

A balanced pair in an abelian category yields a Quillen equivalence between model structures on its two classes of chain complexes when suitable conditions hold.

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

The paper sets out conditions under which the chain homotopy categories K(X) and K(Y) for a balanced pair (X, Y) become triangulated equivalent. It does so by placing model structures on the categories of chain complexes with objects drawn from X and from Y, then showing that a Quillen equivalence between those model categories produces the desired triangulated equivalence of homotopy categories. A reader would care because the result supplies a uniform way to obtain equivalences between homotopy categories attached to different classes of objects, and the authors apply it to cotorsion triples as well as to Gorenstein projective, Gorenstein injective, pure projective and pure injective modules.

Core claim

For a balanced pair (X, Y) in an abelian category, the authors give conditions that guarantee a Quillen equivalence between the model categories on the chain complexes Ch(X) and Ch(Y). The homotopy categories of these model categories are precisely the chain homotopy categories K(X) and K(Y), so the Quillen equivalence implies that K(X) and K(Y) are triangulated equivalent. The same conditions are shown to hold in several concrete settings, including cotorsion triples and the pairs formed by Gorenstein projective with Gorenstein injective modules, and by pure projective with pure injective objects.

What carries the argument

The Quillen equivalence between the model categories placed on Ch(X) and Ch(Y) that is induced by the balanced pair (X, Y).

If this is right

  • Whenever the conditions on the balanced pair hold, the triangulated categories K(X) and K(Y) are equivalent.
  • The equivalence applies directly to any cotorsion triple, producing a triangulated equivalence between the corresponding chain homotopy categories.
  • The homotopy categories of Gorenstein projective modules and Gorenstein injective modules are triangulated equivalent under the given conditions.
  • The homotopy categories of pure projective objects and pure injective objects are triangulated equivalent under the given conditions.

Where Pith is reading between the lines

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

  • The construction may allow properties proved for one side of the balanced pair to be transferred to the other side via the equivalence of homotopy categories.
  • Similar model-categorical arguments could be attempted for other pairs of classes that are not necessarily balanced but still admit compatible resolutions.
  • The result supplies a model-category route to known triangulated equivalences in Gorenstein homological algebra that were previously obtained by different means.

Load-bearing premise

The chain homotopy categories K(X) and K(Y) can be realized as the homotopy categories of model structures on the categories of chain complexes whose objects lie in X and in Y respectively.

What would settle it

An explicit balanced pair satisfying the paper's stated conditions for which the associated model categories on Ch(X) and Ch(Y) are not Quillen equivalent.

read the original abstract

For a balanced pair $(\mathcal{X},\mathcal{Y})$ in an abelian category, we investigate when the chain homotopy categories ${\bf K}(\mathcal{X})$ and ${\bf K}(\mathcal{Y})$ are triangulated equivalent. To this end, we realize these chain homotopy categories as homotopy categories of certain model categories and give conditions that ensure the existence of a Quillen equivalence between the model categories in question. We further give applications to cotorsion triples, Gorenstein projective and Gorenstein injective modules, as well as pure projective and pure injective objects.

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 / 3 minor

Summary. The manuscript studies balanced pairs (X, Y) in an abelian category and gives conditions ensuring that the chain homotopy categories K(X) and K(Y) are triangulated equivalent. It equips the categories of chain complexes valued in X and in Y with model structures whose homotopy categories recover K(X) and K(Y), then identifies conditions (involving cotorsion pairs and compatibility with the abelian structure) under which the induced adjunction is a Quillen equivalence. Applications are supplied to cotorsion triples, Gorenstein projective and injective modules, and pure projective and pure injective objects.

Significance. If the stated conditions suffice, the work supplies a model-categorical route to triangulated equivalences of homotopy categories arising from balanced pairs. The explicit specialization to Gorenstein and pure-injective settings provides concrete illustrations that may connect existing results in homological algebra.

minor comments (3)
  1. [Abstract] The abstract states that conditions are given for the Quillen equivalence, but does not name the precise hypotheses (e.g., the cotorsion-pair compatibility condition); adding a one-sentence summary of the main theorem would improve readability.
  2. [Section 2] Notation for the model structures on Ch(X) and Ch(Y) is introduced without an early global table or diagram; a short summary table of the cofibrations, fibrations, and weak equivalences would aid the reader.
  3. [Section 5] In the applications section the specialization to Gorenstein projective modules is stated to follow by direct substitution, but the verification that the balanced-pair hypotheses hold in that case is only sketched; a one-paragraph check would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal at this stage. We remain available to incorporate any minor editorial suggestions in a revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines model structures on chain complexes valued in the classes of a balanced pair (X,Y) so that their homotopy categories recover K(X) and K(Y), then states explicit compatibility conditions (cotorsion pairs, abelian structure) under which the induced adjunction is a Quillen equivalence. All verifications proceed by direct checking of the model-category axioms via lifting properties and by confirming that derived unit and counit are weak equivalences on appropriate objects; these steps rely on the external definition of balanced pairs and standard homological algebra rather than any self-referential fit, renaming, or load-bearing self-citation. Applications to Gorenstein and pure-injective settings are obtained by specialization of the same general conditions. No equation or claim reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical axioms of abelian categories and model categories; no free parameters or new entities introduced based on the abstract.

axioms (1)
  • standard math Standard axioms of category theory and model categories
    The paper relies on the existence of model structures on chain complexes.

pith-pipeline@v0.9.0 · 5618 in / 1297 out tokens · 74723 ms · 2026-05-23T02:10:18.884693+00:00 · methodology

discussion (0)

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Reference graph

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