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Functor bridges n+1-term subcategories and complexes

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2026-07-09 01:01 UTC pith:S4VYPSAD

load-bearing objection Generalizes Yang's 2-term functor to (n+1)-term complexes; the construction is sound but one key composition is asserted rather than displayed. the 1 major comments →

arxiv 2607.06960 v1 pith:S4VYPSAD submitted 2026-07-08 math.RT math.CT

From (n+1)-term subcategories to (n+1)-term complexes

Zhaotai Zhang , Yu Zhou , Bin Zhu This is my paper
classification math.RT math.CT MSC 18G8016E3518G05
keywords extriangulated categoryn-rigid subcategorysilting subcategorycluster tiltingCalabi-Yau categoryhomotopy categorymutationAuslander category
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper constructs an explicit functor P from the (n+1)-term subcategory of a triangulated category generated by an n-rigid subcategory M to the category of (n+1)-term complexes over M, generalizing a previously known construction for the n=1 case. The functor P sends each object, built from n+1 consecutive shifts of M, to a complex of length n+1 built from M, and is shown to respect the extriangulated (generalized exact) structures on both sides. The main theorem proves that P is full if and only if a natural map on extensions is injective, if and only if the groups Hom(Σ^i M, M) vanish for 1 ≤ i ≤ n−1. When any of these equivalent conditions holds, P becomes dense, the extension map becomes a natural isomorphism, and P descends to an extriangle equivalence between the quotient of the source category by a specific ideal and the target complex category. The paper further shows that this equivalence preserves silting subcategories and their mutations, and applies the results to n-cluster tilting subcategories and to n-cluster tilting objects in (n+1)-Calabi-Yau categories, obtaining bijections between cluster tilting objects and silting or tilting complexes.

Core claim

The central discovery is the trifold equivalence: fullness of P, injectivity of the extension map Ω, and the vanishing of Hom(Σ^i M, M) for intermediate shifts 1 ≤ i ≤ n−1. This vanishing condition, which is weaker than full n-rigidity (it excludes only the extreme i = n), is the precise obstruction that determines whether the functor from (n+1)-term subcategories to (n+1)-term complexes is an equivalence. The functor P itself is constructed by lifting objects and morphisms through a Frobenius exact category, using presentations of length n+1 and the horseshoe lemma to build chain maps between complexes, then passing to the stable category. The natural isomorphism Γ between P∘Σ and Σ∘P on a子

What carries the argument

The functor P is built from eM-presentations of length n+1 in a Frobenius exact category F whose stable category is T. Each object X in pr^{n+1}_T(M) is resolved by n+1 conflations using objects from M, producing a complex C•(X) in degrees −n to 0. Morphisms are lifted to chain maps via vanishing of Ext^1(M, Y) for appropriate objects, and null-homotopies are controlled by factoring through projective-injective objects. The extension map Ω is defined by factoring extensions through Σ of objects in pr^n_T(M) and composing with Γ, a natural isomorphism P∘Σ ≅ Σ∘P constructed from the standard triangle structure of the stable category.

Load-bearing premise

The explicit construction of the functor P requires the triangulated category T to be algebraic (the stable category of a Frobenius exact category), so that morphisms can be lifted to the exact category and chain maps can be built using the horseshoe lemma. The main theorem is then stated for any reduced (n−1)-Auslander extriangulated category, relying on a cited classification result that every algebraic such category arises from the construction. If that classification hasg

What would settle it

If a reduced (n−1)-Auslander extriangulated category exists that does not arise from an n-rigid subcategory of an algebraic triangulated category, the functor P may not be constructible for that category, limiting the scope of Theorem 5.14.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • When M is n-cluster tilting and the vanishing condition holds, the entire triangulated category T is equivalent (modulo an ideal) to (n+1)-term complexes over M, providing a higher-dimensional analogue of classical tilting theory.
  • In (n+1)-Calabi-Yau categories, the functor gives a mutation-compatible bijection between 2-term n-cluster tilting objects and 2-term silting complexes over the endomorphism algebra, extending the support τ-tilting correspondence to arbitrary n.
  • The equivalence preserves silting subcategories and their left/right mutations, meaning the combinatorial mutation structure on one side is faithfully mirrored on the other.
  • When Σ^{n+1}M = M, the functor P becomes a genuine equivalence of additive categories, endowing K[-n,0](M) with a triangulated structure inherited from T.

Where Pith is reading between the lines

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

  • The vanishing condition Hom(Σ^i M, M) = 0 for 1 ≤ i ≤ n−1 is strictly weaker than n-rigidity (which includes i = n), suggesting that the 'boundary' extension group Hom(Σ^n M, M) plays a different role—it controls the ideal [Σ^n M, M] being quotiented, rather than obstructing the equivalence.
  • The construction likely extends to non-algebraic triangulated categories if one can find an alternative to the Frobenius lift, since the main theorem is stated for general reduced (n−1)-Auslander extriangulated categories, but the explicit construction of P currently depends on the algebraic hypothesis.
  • The mutation compatibility of the silting bijection suggests that exchange graphs of silting subcategories on both sides are isomorphic, which could provide a combinatorial tool for computing mutation classes in settings where one side is more tractable.

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

1 major / 7 minor

Summary. This paper constructs an extriangulated functor (P, Ω) from the (n+1)-term subcategory pr_T^{n+1}(M) of an algebraic triangulated category T to the category K[-n,0](M) of (n+1)-term complexes, generalizing Yang's n=1 construction. The functor P restricts to the identity on M. The main result (Theorem 1.2 / Theorems 5.14, 5.17, 5.20) establishes that P is full if and only if Ω is injective if and only if Hom_C(Σ^i M, M) = 0 for 1 ≤ i ≤ n−1, and under these conditions P induces an extriangle equivalence C/[Σ^n M, M] ≃ K[-n,0](M) plus a mutation-compatible bijection on silting subcategories. Applications to n-cluster tilting subcategories and (n+1)-Calabi-Yau categories are given in Section 6.

Significance. The paper makes a substantial contribution by extending the functorial bridge between (n+1)-term subcategories and (n+1)-term complexes from the classical n=1 case to arbitrary n. The construction of P in Section 3 is parameter-free (given M and T, the functor is determined up to natural isomorphism by Proposition 3.9), and the equivalent conditions in Theorem 5.14 are logically derived rather than circularly imposed. The n=1 case recovers Yang's result, providing a consistency check. The applications to silting bijections and cluster tilting in Calabi-Yau categories are concrete and falsifiable. The proofs proceed by careful induction with explicit chain maps.

major comments (1)
  1. Proof of Theorem 4.4 (end of p. 23): The final step of the proof asserts that the composition of the three explicitly written chain maps — (3.14), (3.12), and (4.7) — equals the chain map shown in (4.5), verified by 'a direct check.' This composition involves alternating signs (−1)^i and index shifts across three maps, and it is the load-bearing step establishing that (P, Ω) is an extriangulated functor. If any sign or degree alignment is off, Ω fails to send extriangles to triangles, invalidating Theorem 1.1, Theorem 5.21, and the Section 6 applications. The n=1 case provides one check, but the first genuinely new case n=2 has no prior verification. The authors should either display the term-by-term composition or, at minimum, spell out the verification for one representative degree index to confirm the sign convention is consistent across all three maps.
minor comments (7)
  1. p. 4, Convention: 'For any two subcategories X and Y of a triangulated category T, we denote by X*Y the subcategory of T consisting of the objects Z such that there exists a triangle X → Z → Y → ΣX with X ∈ X and Y ∈ Y.' This should specify X ∈ X (an object) rather than X ∈ X (the subcategory), for clarity.
  2. p. 6, line 5: 'natrual' should be 'natural' (appears in the proof of Theorem 4.4: 'since Ω is a natrual transformation').
  3. p. 23, line 2: 'ΣeP(eδ)◦eΓ_{X1}◦eP(eγ_X^0)' — the notation with the 'e' prefix is used consistently for lifts to F, but a brief reminder at this point that 'e' denotes the lifted version would aid readability, as this is a dense passage.
  4. Example 2.17 (p. 10): The AR quiver is drawn with specific notation (P_i, S_i) but the relationship between the 2-cluster category structure and the specific triangles listed could benefit from one sentence explaining why M = add(P_1 ⊕ ΣP_2) is 2-rigid.
  5. p. 37, Corollary 6.6: 'Hom_T(ΣM, M) = 0 for all 1 ≤ i ≤ n−1' should read 'Hom_T(Σ^i M, M) = 0 for all 1 ≤ i ≤ n−1' (missing exponent on Σ).
  6. The reference [36] to Silberberg's forthcoming work is cited as 'In preparation.' If available by the time of revision, a more complete reference or arXiv link would be appropriate.
  7. Section 5 is dense; a brief roadmap paragraph at the start of Section 5 outlining the logical flow (Proposition 5.3 → Lemma 5.9 → Lemma 5.11 → Theorem 5.14 → Proposition 5.16 → Theorems 5.17, 5.20) would help the reader navigate the induction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestion regarding the sign verification in the proof of Theorem 4.4. We agree that this step is load-bearing and that the verification should be made explicit rather than left as a 'direct check.' We will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: Proof of Theorem 4.4 (end of p. 23): The final step of the proof asserts that the composition of the three explicitly written chain maps — (3.14), (3.12), and (4.7) — equals the chain map shown in (4.5), verified by 'a direct check.' This composition involves alternating signs (−1)^i and index shifts across three maps, and it is the load-bearing step establishing that (P, Ω) is an extriangulated functor. If any sign or degree alignment is off, Ω fails to send extriangles to triangles, invalidating Theorem 1.1, Theorem 5.21, and the Section 6 applications. The n=1 case provides one check, but the first genuinely new case n=2 has no prior verification. The authors should either display the term-by-term composition or, at minimum, spell out the verification for one representative degree index to confirm the sign convention is consistent across all three maps.

    Authors: We fully agree with the referee that this step deserves explicit verification. The composition of the three chain maps (3.14), (3.12), and (4.7) is indeed the crux of the proof that (P, Ω) is an extriangulated functor, and the alternating signs (−1)^i together with the degree shifts make a terse 'direct check' inadequate for a load-bearing argument. We will revise the manuscript to include an explicit term-by-term verification of the composition. Specifically, we will display the composition at a representative degree index i (for 0 ≤ i ≤ n−1), showing how the signs (−1)^{i+1} from (4.7), the identity maps and sign (−1) factors from (3.12), and the signs (−1)^n, (−1)^{n−1}, … from (3.14) combine to produce exactly the entries of the chain map (4.5). We will also verify the boundary cases i = 0 and i = n−1 separately, as these involve the terms I(X₁) and the zero objects where the degree alignment is most delicate. This will confirm the sign convention is consistent across all three maps for arbitrary n, not just n = 1. revision: yes

Circularity Check

1 steps flagged

No significant circularity; one self-citation to prior classification result is not load-bearing for the main theorem's logical derivation.

specific steps
  1. self citation load bearing [Introduction, paragraph after Theorem 1.1 (p. 3)]
    "Conversely, as shown in [13, Theorem 3.7] every algebraic reduced (n−1)-Auslander extriangulated category arises in this way up to extriangulated equivalence. Consequently, given any algebraic reduced (n−1)-Auslander extriangulated category (C,E), Theorem 1.1 yields an extriangulated functor from C to K[−n,0](M) that restricts to the identity functor on M"

    Reference [13] is by X. Chen (a different author, not among Zhang/Zhou/Zhu), so this is not a self-citation in the strict sense. Moreover, [13, Theorem 3.7] is invoked only to extend the applicability of Theorem 1.1 from pr^{n+1}_T(M) to general algebraic reduced (n−1)-Auslander categories. The main result Theorem 5.14 is a conditional statement ('Let (P,Ω) be an extriangulated functor...') whose proof uses only extriangulated category axioms, not [13]. The citation concerns scope of applicability, not logical load-bearing for the equivalence proof itself.

full rationale

The paper's central derivation chain is largely self-contained. The functor P is constructed parameter-free from M and T (Construction 3.7, Proposition 3.9), with uniqueness up to natural isomorphism established independently. The equivalent conditions in Theorem 5.14 — (i) P full, (ii) Ω injective, (iii) Hom_C(Σ^i M, M) = 0 — are logically derived via a chain of lemmas (5.9–5.16) using extriangulated category axioms, not circularly defined. Condition (iii) is an independent algebraic vanishing condition on M, not a restatement of (i) or (ii). The proof of (i)⇒(iii) uses fullness of P plus Lemma 5.9 (P(Σ^{i+1}M) ≅ Σ^{i+1}P(M)) to force Ω((ε_M^i)#(f)) = 0, then fullness to lift a splitting from K^b(M) back to C. The proof of (ii)⇒(iii) uses injectivity of Ω plus the same vanishing in K^b(M). The reverse implications (iii)⇒(i),(ii) proceed through Proposition 5.16, which establishes bijections via induction on projective dimension using the five-lemma-style Lemma 5.13. The one external citation that could be considered load-bearing for applicability — [13, Theorem 3.7] by Chen — is not a self-citation (Chen is not an author of this paper) and concerns only the scope of categories to which Theorem 5.14 applies, not the correctness of the theorem itself. The n=1 case recovering Yang's [37, Theorem 1.1] provides an external consistency check. The 'direct check' at the end of Theorem 4.4's proof is a correctness risk (potential sign/indexing error), not a circularity issue. Overall, the derivation is self-contained against external benchmarks with only minor external citations for scope.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are introduced; the functor P is determined up to natural isomorphism by the input data (M, T). No new mathematical entities (particles, forces, dimensions) are postulated. The axioms are standard domain assumptions (algebraic triangulated category, n-rigidity) or standard mathematical axioms (extriangulated category axioms, octahedral axiom). The classification result [13, Theorem 3.7] is a cited theorem used as a black box.

axioms (5)
  • domain assumption T is an algebraic triangulated category (stable category of a Frobenius exact category F)
    Used throughout Section 3 to lift constructions from T to F. Required for the definition of P (Definition 3.12) and the proof that (P,Ω) is an extriangulated functor (Theorem 4.4).
  • domain assumption M is an n-rigid subcategory of T closed under direct summands
    Used to define the extriangulated structure on pr^{n+1}_T(M) (Proposition 2.14) and to establish the vanishing in equation (2.7) which is used throughout.
  • domain assumption Every algebraic reduced (n−1)-Auslander extriangulated category arises as pr^{n+1}_T(M) up to extriangulated equivalence [13, Theorem 3.7]
    Cited in the introduction to justify applying Theorem 1.1 to general reduced (n−1)-Auslander extriangulated categories in Theorem 1.2. This is a result from a cited reference, not proved here.
  • standard math Standard axioms of extriangulated categories [32, Definition 2.12]
    Used throughout, e.g., ET4 axiom applied in Proposition 5.3 and Proposition 5.6.
  • standard math Octahedral axiom for triangulated categories
    Used in Lemma 2.11 and Lemma 2.13 to establish closure under extensions.

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Pith. "Pith review of From $(n+1)$-term subcategories to $(n+1)$-term complexes." pith.science (2026). https://pith.science/paper/S4VYPSAD

@misc{pith2026260706960,
  author       = {Pith},
  title        = {Pith review of: From $(n+1)$-term subcategories to $(n+1)$-term complexes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4VYPSAD}},
  note         = {Machine review of arXiv:2607.06960}
}
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read the original abstract

Let $n$ be a positive integer. Given an $n$-rigid subcategory $\mathcal{M}$ of an algebraic triangulated category $\mathcal{T}$, we explicitly construct an extriangulated functor from the $(n+1)$-term subcategory of $\mathcal{T}$ generated by $\mathcal{M}$ to the full subcategory of $(n+1)$-term complexes in the bounded homotopy category $K^b(\mathcal{M})$, which restricts to the identity on $\mathcal{M}$. In the broader context of reduced $(n-1)$-Auslander extriangulated categories, we provide necessary and sufficient conditions for such a functor to be full, in which case it induces an equivalence of extriangulated categories modulo a certain ideal. Furthermore, we establish a mutation-compatible bijection between the silting subcategories of these categories. Finally, we apply these results to $n$-cluster tilting subcategories and $n$-cluster tilting objects in $(n+1)$-Calabi-Yau categories.

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This paper was first reviewed by glm-5.2 on July 9, 2026.