Extriangulated factorization systems, $s$-torsion pairs and recollements

Haicheng Zhang; Yan Xu; Zhiwei Zhu

REVIEW 3 major objections 6 minor 15 references

Extriangulated factorization systems, $s$-torsion pairs and recollements

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves a bijection between s-torsion pairs and inflation factorization systems in extriangulated categories with negative first extensions.

desk verdict A clean unification of torsion-pair/t-structure bijections in extriangulated categories, with the main proof leaning on one cited proposition that the authors should state explicitly. read the letter →

arxiv 2507.04220 v1 pith:QVHIYJ5X submitted 2025-07-06 math.CT

Yan Xu , Haicheng Zhang , Zhiwei Zhu This is my paper · ORCID

classification math.CT MSC 18E4018E3018E10

keywords extriangulatedcategoriess-torsionpairsinflationfactorizationsystemsnegativefirstextensionsrecollementst-structurestorsionsiltingcomplexes

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 is a common setting that generalizes abelian and triangulated categories, and this paper builds a two-way dictionary inside it. One side is an $s$-torsion pair, two subcategories $(\mathcal{T},\mathcal{F})$ such that every object decomposes through an extension triangle with one object in each part and such that ordinary morphisms and negative-first-extension groups between the two classes vanish. The other side is an inflation factorization system, a pair of classes of inflations that factor every inflation into a left-class map followed by a right-class map and that are mutually orthogonal. The main theorem states that sending $(\mathcal{T},\mathcal{F})$ to the inflations with cone in $\mathcal{T}$ or $\mathcal{F}$, and reversing by taking cones, gives mutually inverse bijections. The paper also proves the dual statement for deflations and shows the bijections glue under recollements, so a structure on the sides of a recollement determines one on the middle category. If correct, this unifies the classical torsion-pair and t-structure bijections in one general framework.

What carries the argument

The load-bearing object is the pair of operators $\operatorname{Infl}$ and $\operatorname{Cone}$, which translate between two types of data. An $s$-torsion pair $(\mathcal{T},\mathcal{F})$ consists of subcategories with $\mathcal{C}=\mathcal{T}*\mathcal{F}$, $\mathcal{C}(\mathcal{T},\mathcal{F})=0$, and $E^{-1}(\mathcal{T},\mathcal{F})=0$; an inflation factorization system $(\mathcal{L},\mathcal{R})$ is a pair of classes of inflations that factor every inflation and satisfy $\mathcal{L}={}^{\perp}\mathcal{R}$ and $\mathcal{L}^{\perp}=\mathcal{R}$. The proof's engine is showing that the two orthogonality conditions match exactly, with the reverse inclusion $\operatorname{Infl}\mathcal{T} \supseteq {}^{\perp}\operatorname{Infl}\mathcal{F}$ supplied by the cited proposition on cones.

What would settle it

Look for an extriangulated category with negative first extensions and an $s$-torsion pair $(\mathcal{T},\mathcal{F})$ containing an inflation $h$ with $\mathcal{C}(\operatorname{cone}(h), F)=0$ and $E^{-1}(\operatorname{cone}(h), F)=0$ for all $F\in\mathcal{F}$, yet $\operatorname{cone}(h)\notin\mathcal{T}$. Such an example would make the equality $\operatorname{Infl}\mathcal{T}={}^{\perp}\operatorname{Infl}\mathcal{F}$ fail and disprove Theorem 3.1.

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Extended reading notes

Core claim

The central claim is Theorem 3.1: in any extriangulated category with negative first extensions, the maps $(\mathcal{T},\mathcal{F}) \mapsto (\operatorname{Infl}\mathcal{T}, \operatorname{Infl}\mathcal{F})$ and $(\mathcal{L},\mathcal{R}) \mapsto (\operatorname{Cone}\mathcal{L}, \operatorname{Cone}\mathcal{R})$ are mutually inverse bijections between $s$-torsion pairs and inflation factorization systems. Orthogonality of two inflations $l,r$ is defined by simultaneous vanishing of the ordinary hom-group and the negative first extension group between their cones. The proof factors an arbitrary inflation through the $s$-torsion decomposition of its cone using the extriangulated axiom (ET4)$^{\mathrm{op}}$, and uses a cited proposition asserting that zero hom-groups force the cone into $\mathcal{T}$. The dual Theorem 3.2 gives the corresponding bijection for deflation factorization systems, and the corollaries recover the known bijections between torsion pairs and monomorphism factorization systems in abelian categories and between t-structures and inflation factorization systems in triangulated categories.

Load-bearing premise

The bijection depends on a previously published proposition which says that an inflation whose cone has zero morphism space and zero negative-first-extension space to every object of $\mathcal{F}$ must have its cone in $\mathcal{T}$; that proposition is not proved here, and the reverse direction of the bijection collapses if it is false.

Editorial extensions

If this is right

In any extriangulated category with negative first extensions, every statement about $s$-torsion pairs can be translated into a statement about inflation factorization systems, and conversely.
The bijection recovers the classical correspondence between torsion pairs in abelian categories and monomorphism or epimorphism factorization systems, and between t-structures and factorization systems in triangulated categories.
Given a recollement with balanced negative first extensions, compatible $s$-torsion pairs on the outer categories determine an $s$-torsion pair on the middle, and the same gluing works for factorization systems.
Silting complexes in derived categories yield explicit examples of extriangulated factorization systems, as demonstrated for a 2-extended module category of a path algebra.

Reading between the lines

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

The bijection likely extends to a notion of extriangulated torsion theory, defined by requiring both classes in a factorization system to have the 3-for-2 property; this would give a normal-torsion-theory analogue in the extriangulated setting.
Because the proof is constructive, it may be possible to compute the factorization system from a silting complex and vice versa, giving a practical tool in representation theory.
A natural stress test is to see whether the bijection survives when the negative first extension structure is replaced by higher negative extensions; new orthogonality conditions might enter and change the dictionary.
The gluing theorem suggests that recollement compatibility could be formulated directly as a descent condition on factorization systems, bypassing s-torsion pairs entirely.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take on arXiv:2507.04220.

The paper does something genuinely useful: it defines extriangulated factorization systems (inflation and deflation versions) and proves a bijection between s-torsion pairs and inflation factorization systems in any extriangulated category with negative first extensions. This really does unify the Rosicky-Tholen bijection for torsion pairs in abelian categories and the Loregian-Virili bijection for t-structures in triangulated categories, and the categorical machinery is the right level of generality. The gluing theorem under recollements (Theorem 4.4) is a natural and useful extension, and Example 3.8 with the 2-extended module category is a helpful concrete illustration.

The main proof is structurally sound: the factorization step uses ET4^op, and the orthogonality in one direction follows directly from the s-torsion pair conditions. The soft spot is the converse inclusion in Theorem 3.1: to show that h in ^\perp Infl F has cone(h) in T, the paper invokes [1, Proposition 3.2] without stating the proposition or checking that its hypotheses match. That is not a small step; it is exactly where the E^{-1} orthogonality does its work. [1] is published, so this is a delegation rather than a gap, but the paper would be much more self-contained and easier to referee if the authors stated the proposition and gave a one-paragraph proof in their setting. The dual inclusion is dismissed with 'similarly', which is fine but worth a few lines.

Two smaller issues. First, the text says nothing about the relation to [9], which is about admissible weak factorization systems on extriangulated categories; a comparison remark is needed, because the reader will legitimately wonder whether the inflation factorization systems here overlap with that work. Second, the commutative diagram in the proof of Theorem 3.1 appears corrupted in the arXiv text; that is likely a rendering artifact, but it should be fixed in a revision.

In my view the central bijection holds up; the result is a proper generalization and the paper is honest about what it depends on. I would send this to a serious referee. The referee should ask for (a) a statement/proof of the cited proposition in context, and (b) a discussion of [9]. The paper is aimed at researchers in representation theory and category theory who work with extriangulated categories, torsion pairs, and recollements. For that audience it is a useful contribution.

Referee Report

3 major / 6 minor

Summary. The paper introduces inflation factorization systems in extriangulated categories with negative first extensions, consisting of two classes of inflations closed under a cone-based orthogonality condition and a factorization axiom. The main result (Theorem 3.1) asserts a bijection between s-torsion pairs and such inflation factorization systems; Theorem 3.2 is the dual statement for deflation factorization systems. Corollaries recover bijections for torsion pairs in abelian categories and t-structures in triangulated categories. Section 4 defines balanced negative first extensions and proves gluing theorems for s-torsion pairs and factorization systems under recollements of extriangulated categories, with an example in 2-extended module categories.

Significance. If fully established, the bijection of Theorem 3.1 provides a common framework for the classical bijection between torsion theories and factorization systems in abelian categories and for the bijection between t-structures and triangulated factorization systems. The paper is genuinely synthetic: the definitions of inflation factorization systems and cone/cocone classes are natural, the corollaries are immediate, and the worked example in Example 3.8 illustrates the construction. The main weaknesses are that the proof of the central theorem delegates the key orthogonality step to an unstated external proposition and that the gluing section relies on an incompletely justified lemma and on external theorems; these points must be repaired before the contribution is fully self-contained.

major comments (3)

[Section 3, proof of Theorem 3.1] The inclusion ⊥Infl F ⊆ Infl T is the decisive step of the bijection and is justified only by the bare citation '[1, Proposition 3.2]' after observing that C(cone(h),F)=0 and E^{-1}(cone(h),F)=0 for all F∈F. The proposition is not stated, so the reader cannot verify that its hypotheses are satisfied in the present setting. The dual inclusion (Infl T)^⊥ ⊆ Infl F is dismissed with 'Similarly'; since Definition 2.5 requires both L = ⊥R and L^⊥ = R, both inclusions are load-bearing for the bijection. Please state the needed proposition with its hypotheses and prove both inclusions in full.
[Section 4, Lemma 4.3] The proof draws a commutative diagram of exact sequences whose left vertical map is the desired isomorphism E^{-1}_A(FX,Y) ≅ E^{-1}_B(X,GY), but that isomorphism is exactly what the lemma is supposed to establish. The vertical maps on the right are adjunction isomorphisms, yet the paper does not show that the connecting maps in the two exact sequences are compatible with these adjunction isomorphisms; this requires a naturality statement for the negative-first-extension transformations under the functor G and its adjoint. Without an explicit verification that the diagram commutes, the Five-Lemma argument is circular.
[Section 4, Theorem 4.4] The proof invokes Lemma 4.3 twice but does not verify the lemma's hypothesis that the relevant source category 'has enough projectives' in each application; this hypothesis is not among the theorem's assumptions. The equality B = T * F is also not proved, being deferred to '[6, Theorem 3.4]' with no explanation of why that result applies to the present balanced-negative-first-extensions setting. Please either add the missing hypotheses and reasoning or reproduce the argument so that the gluing theorem is self-contained.

minor comments (6)

[Section 4, proof of Theorem 4.4] The displayed isomorphisms B(i_* i^* T, F) ≅ B(i^* T, i^! F) and B(j_! j^* T, F) ≅ B(j^* T, j^* F) should be Hom_A(i^*T, i^!F) and Hom_C(j^*T, j^*F), respectively; as printed they misstate which Hom-category is involved.
[Section 3, proof of Theorem 3.1] The commutative diagram of conflations obtained by (ET4)^op is garbled in the typeset version; please redraw it so that the objects and morphisms can be checked.
[Abstract and Definition 2.5] The abstract and introduction call the new notion an 'extriangulated factorization system,' but Definition 2.5 defines an 'inflation factorization system,' which factorizes only inflations. Please qualify the terminology in the abstract and introduction to avoid suggesting a factorization system on all morphisms.
[Example 3.8] Example 3.8 is difficult to verify as printed: the Auslander–Reiten quiver and the morphisms f, l, r are not legible in the provided typesetting, and the sets X and Y are displayed in a way that makes the factorization f = rl hard to check. Please typeset the example with labeled morphisms.
[References] Reference [10] (Mac Lane, 'Duality for groups') does not appear to be cited anywhere in the body of the paper; please remove it or cite it in the relevant discussion.
[Section 4, Theorems 4.5 and 4.6] Theorems 4.5 and 4.6 state 'balanced negative extensions' while Definition 4.2 and Lemma 4.3 use 'balanced negative first extensions'; please make the terminology uniform.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central bijection is proved from the definitions plus one external cited proposition; the only self-citation is background material in the gluing section.

full rationale

The derivation is not circular. Theorem 3.1 proves that (T,F) ↦ (Infl T, Infl F) is an inflation factorization system for any s-torsion pair. The nontrivial inclusion ⊥Infl F ⊆ Infl T is delegated to [1, Proposition 3.2], an external result of Adachi–Enomoto–Tsukamoto, not to an assumption of this paper; the paper's definitions do not encode the conclusion. The converse direction (L,R) ↦ (Cone L, Cone R) is checked directly from the s-torsion pair conditions and the orthogonality axioms, using the factorization of the inflation 0 → X to obtain Cone(L) ∗ Cone(R) = C and using L ⊥ R to get the vanishing of Hom and E^{-1}. Short inclusions such as Cone(Infl T) = T and Infl(Cone L) ⊆ L are proved from the E-triangles 0 → T → T and from L = ⊥R, respectively. The only self-citation, [14] by Wang–Wei–Zhang, is used solely in Section 4 for background properties of recollements of extriangulated categories; it does not carry the central bijection. The reliance on the unproved external proposition [1, Prop. 3.2] and the omitted dual inclusion are rigor concerns, not circularity: a gap or an external dependency does not make the conclusion equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper's central claims rest on the framework of extriangulated categories with negative first extensions and on several external results that are cited but not proved. The only genuinely new entities are the factorization systems and the balanced negative first extensions condition; both are defined to make the theorems work, but the main bijection gives them content. No free parameters are involved since this is pure mathematics.

assumptions (6)

domain assumption Existence of a negative first extension structure on the extriangulated category (Definition 2.2).
The theory of s-torsion pairs and the orthogonality condition in Definition 2.4 both use E^{-1}; the main theorems are only stated for categories with this structure, imported from [1].
standard math The axioms (ET1)-(ET4), (ET3)op, (ET4)op of extriangulated categories from [12, Definition 2.12] hold.
These are the foundational axioms of extriangulated categories, taken as background from Nakaoka-Palu.
domain assumption Balanced negative first extensions (Definition 4.2), namely E^{-1}(-, P(C)) = 0.
This new condition is used in Lemma 4.3 and is assumed in Theorems 4.4-4.6. The paper only notes that exact and triangulated categories satisfy it.
standard math [1, Proposition 3.2]: a characterization of s-torsion pairs used to pass from orthogonality of cones to membership in T.
Invoked in the proof of Theorem 3.1 to show that an inflation whose cone is orthogonal to F belongs to Infl T. It is not proved in the paper.
standard math [6, Theorem 3.4]: gluing of torsion pairs under recollements, used to establish B = T*F in Theorem 4.4.
The proof of Theorem 4.4 explicitly says 'The proof of B = T*F is the same as given in the proof of [6, Theorem 3.4]'.
standard math Recollement properties from [14, Lemma 3.3 and Proposition 3.4], including exactness and adjunction identities.
Used repeatedly in the proofs of Theorems 4.4 and 4.5 to transfer orthogonality and extension conditions between A, B, and C.

invented entities (2)

Inflation factorization system / extriangulated factorization system independent evidence
purpose: To encode s-torsion pairs as pairs of morphism classes; the central bijection Theorem 3.1 is the main result.
The concept supports a bijection with known structures (torsion pairs, t-structures) and yields a concrete example in 2-mod A (Example 3.8), giving it independent mathematical content beyond the definition.
Balanced negative first extensions
purpose: To make Lemma 4.3 work, enabling the gluing of s-torsion pairs and factorization systems under recollements.
This is a new technical condition introduced in Definition 4.2. The paper only shows that exact and triangulated categories satisfy it; it provides no external falsifiable predictions.

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Cite this review

Pith. "Pith review of Extriangulated factorization systems, $s$-torsion pairs and recollements." pith.science (2026). https://pith.science/paper/QVHIYJ5X

@misc{pith2026250704220,
  author       = {Pith},
  title        = {Pith review of: Extriangulated factorization systems, $s$-torsion pairs and recollements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVHIYJ5X}},
  note         = {Machine review of arXiv:2507.04220}
}

abstract

We introduce extriangulated factorization systems in extriangulated categories and show that there exists a bijection between $s$-torsion pairs and extriangulated factorization systems. We also consider the gluing of $s$-torsion pairs and extriangulated factorization systems under recollements of extriangulated categories.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages

[1]

Adachi, H

T. Adachi, H. Enomoto, M. Tsukamoto, Intervals of s-torsion pairs in extriangulated categories with negative first extensions, Math. Proc. Cambridge Philos. Soc. 174(3) (2023), 451–469

work page 2023
[9]

Y. Ma, H. You, D. Zhang, P. Zhou, Admissible weak factorization systems on extriangulated cate- gories, J. Algebra Appl. (2025), online

work page 2025
[2]

Beilinson, J

A.A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, in: Analysis and topology on singular spaces, I. (Luminy, 1981,) Asterisque, vol. 100, Soc. Math. France, Paris (1982), 5–171

work page 1982
[3]

Bekkert, H.A

V. Bekkert, H.A. Merklen, Indecomposables in derived categories of gentle algebras, Algebr. Repre- sent. Theory 6(3) (2003), 285–302

work page 2003
[4]

Cassidy, M

C. Cassidy, M. H´ ebert, G. M. Kelly, Reflective subcategories, localazations and factorization systems, J. Australian Math. Soc. (Series A) 38(3) (1985), 287–329

work page 1985
[5]

Gorsky, H

M. Gorsky, H. Nakaoka, Y. Palu, Positive and negative extensions in extriangulated categories, arXiv:2103.12482 (2021)

arXiv 2021
[6]

J. He, Y. Hu, P. Zhou, Torsion pairs and recollements of extriangulated categories, Comm. Algebra 50(5) (2022), 2018–2036

work page 2022
[7]

Y. Liu, H. Nakaoka, Hearts of twin cotorsion pairs on extriangulated categories, J. Algebra 528 (2019), 96–149

work page 2019

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[8]

Loregian, S

F. Loregian, S. Virili, Triangulated factorization systems and t-structures, J. Algebra 550 (2020), 219–241

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Maclane, Duality for groups, Bull

S. Maclane, Duality for groups, Bull. Amer. Math. Soc. 56(6) (1950), 485–516

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MacPherson, K

R. MacPherson, K. Vilonen, Elementary construction of perverse sheaves, Invent. Math. 84(2) (1986), 403–435

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Nakaoka, Y

H. Nakaoka, Y. Palu, Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. G´ eom. Diff´ er. Cat´ eg. 60(2) (2019), 117–193

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Rosick´ y, W

J. Rosick´ y, W. Tholen, Factorization, fibration and torsion, J. Homotopy Relat. Struct. 2(2) (2007), 295–314

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L. Wang, J. Wei, H. Zhang, Recollements of extriangulated categories, Colloq. Math. 167(2) (2022), 239–259

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Zhou, Tilting theory for extended module categories, arXiv:2411.15473 (2025)

Y. Zhou, Tilting theory for extended module categories, arXiv:2411.15473 (2025). 14 Y. XU, H. ZHANG, Z. ZHU Ministry of Education Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China Email address: swgfeng@outlook.com...

2025 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Adachi, H

T. Adachi, H. Enomoto, M. Tsukamoto, Intervals of s-torsion pairs in extriangulated categories with negative first extensions, Math. Proc. Cambridge Philos. Soc. 174(3) (2023), 451–469

work page 2023

[2] [9]

Y. Ma, H. You, D. Zhang, P. Zhou, Admissible weak factorization systems on extriangulated cate- gories, J. Algebra Appl. (2025), online

work page 2025

[3] [2]

Beilinson, J

A.A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, in: Analysis and topology on singular spaces, I. (Luminy, 1981,) Asterisque, vol. 100, Soc. Math. France, Paris (1982), 5–171

work page 1982

[4] [3]

Bekkert, H.A

V. Bekkert, H.A. Merklen, Indecomposables in derived categories of gentle algebras, Algebr. Repre- sent. Theory 6(3) (2003), 285–302

work page 2003

[5] [4]

Cassidy, M

C. Cassidy, M. H´ ebert, G. M. Kelly, Reflective subcategories, localazations and factorization systems, J. Australian Math. Soc. (Series A) 38(3) (1985), 287–329

work page 1985

[6] [5]

Gorsky, H

M. Gorsky, H. Nakaoka, Y. Palu, Positive and negative extensions in extriangulated categories, arXiv:2103.12482 (2021)

arXiv 2021

[7] [6]

J. He, Y. Hu, P. Zhou, Torsion pairs and recollements of extriangulated categories, Comm. Algebra 50(5) (2022), 2018–2036

work page 2022

[8] [7]

Y. Liu, H. Nakaoka, Hearts of twin cotorsion pairs on extriangulated categories, J. Algebra 528 (2019), 96–149

work page 2019