{"id":"8b13cd99-9b20-4269-bf12-7d23e28c427b","arxiv_id":"2507.04220","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Extriangulated factorization systems are shown to be equivalent to s-torsion pairs, offering a unified framework that recovers classical torsion pairs and t-structures.","lead":"This paper introduces a new concept, extriangulated factorization systems, and proves they correspond exactly to s-torsion pairs in a broad class of categories. The result unifies earlier theorems that linked two different ideas in algebra, providing a common framework that includes abelian torsion pairs and triangulated t-structures.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's converse inclusion relies on [1, Prop. 3.2], which is cited without proof; the proposition's proof is not immediate, so the stated generality of the bijection is not fully verified.","rationale":"The reader's weakest assumption correctly identifies the reliance on [1, Prop. 3.2] as the most load-bearing point in the proof of Theorem 3.1. My stress-test confirms that this step is not merely cosmetic: it is the only place where object-wise orthogonality is converted into membership in the torsion class, and the paper does not supply the proof. The observation about zero deflations is meant to explain why the proposition is not trivial from the decomposition axiom alone. The rest of the proof of Theorem 3.1, and the gluing arguments in Section 4, are mostly routine consequences of the definitions and the cited recollement properties. The theorem is plausible and probably correct, but as written it is a conditional result pending verification of the delegated proposition. Therefore the reader's CONDITIONAL verdict should stand; no change is needed. The separate issue of overlap with [9] does not affect the correctness concern and does not change this assessment.","tokens_in":9638,"tokens_out":47904,"duration_ms":501161,"concrete_test":"Add a complete proof of [1, Proposition 3.2] to the paper, or state its full hypotheses, and verify that the proof does not use idempotent completeness or extension-closure assumptions absent from §2. If a proof requires such assumptions, restate Theorem 3.1 with those assumptions and check whether the bijection still holds in a homotopy category lacking split idempotents.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step in the proof of Theorem 3.1 is the inclusion ⊥Infl F ⊆ Infl T. The argument notes that h ∈ ⊥Infl F gives C(cone(h),F)=0 and E^{-1}(cone(h),F)=0 for every F∈F, and then invokes [1, Proposition 3.2] to conclude cone(h)∈T. This proposition is the entire content of the converse direction: without it, the assignment (T,F)↦(Infl T, Infl F) is not known to produce an inflation factorization system, and the bijection collapses. The paper gives no proof or statement of the hypotheses of [1, Prop. 3.2]. The naive proof (decompose cone(h) as T_X→X→F_X and use Hom(X,F_X)=0 to force the deflation X→F_X to be zero) is not automatic: in extriangulated/triangulated categories a zero deflation need not make the E-triangle split (e.g., A→0→A[1] is a non-split triangle with zero deflation). Showing that the additional E^{-1} condition rules this out is exactly the nontrivial part. The dual inclusion (Infl T)^⊥=Infl F is also omitted as 'similarly'. Thus the central theorem is only as strong as the unstated proof of [1, Prop. 3.2] under the paper's assumptions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9945,"tokens_out":13942,"duration_ms":142413,"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":[{"comment":"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":"Section 3, proof of Theorem 3.1"},{"comment":"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":"Section 4, Lemma 4.3"},{"comment":"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.","section":"Section 4, Theorem 4.4"}],"minor_comments":[{"comment":"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":"Section 4, proof of Theorem 4.4"},{"comment":"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.","section":"Section 3, proof of Theorem 3.1"},{"comment":"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.","section":"Abstract and Definition 2.5"},{"comment":"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.","section":"Example 3.8"},{"comment":"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":"References"},{"comment":"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.","section":"Section 4, Theorems 4.5 and 4.6"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and addresses a natural unification problem. The central idea is promising, and the corollaries and example demonstrate its reach. However, the main bijection is not fully proved within the paper because the key orthogonality step is outsourced to an unstated proposition, and the gluing theorem depends on a lemma whose proof is circular as written. I would not recommend rejection, but the authors should be asked to expand the proof of Theorem 3.1 or state [1, Prop. 3.2] explicitly, and to repair Lemma 4.3 and the missing hypotheses in Theorem 4.4. The typesetting of the commutative diagram and of Example 3.8 should also be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on arXiv:2507.04220.\n\nThe 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.\n\nThe 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.\n\nTwo 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.\n\nIn 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.","headline":"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.","tokens_in":10475,"tokens_out":3445,"would_cite":true,"duration_ms":33824,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18E40","18E30","18E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a bijection between s-torsion pairs and inflation factorization systems in extriangulated categories with negative first extensions.","keywords":["extriangulated categories","s-torsion pairs","inflation factorization systems","negative first extensions","recollements","t-structures","torsion pairs","silting complexes"],"falsifier":"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.","tokens_in":9430,"feed_emoji":"🔗","tokens_out":7677,"duration_ms":79009,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bijection found: s-torsion pairs match factorization systems","feed_subtitle":"In extriangulated categories, every s-torsion pair yields a factorization system and vice versa, unifying known cases.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the definition of s-torsion pairs with negative first extensions and the proposition the proof uses to conclude cone(h) lies in T.","marker":"[1]"},{"why":"introduces triangulated factorization systems and the bijection with t-structures that this paper generalises.","marker":"[8]"},{"why":"introduces extriangulated categories, the ambient structure carrying the whole argument.","marker":"[12]"},{"why":"gives the classical bijection between torsion pairs and normal torsion theories that the new bijection unifies.","marker":"[13]"},{"why":"defines recollements of extriangulated categories, which the gluing theorems in Section 4 use.","marker":"[14]"},{"why":"provides the silting-complex construction used to build the explicit example of an s-torsion pair and its factorization system.","marker":"[15]"}],"fun_headline_variants":["Extriangulated bijection: s-torsion pairs ↔ factorization systems","New bijection links s-torsion pairs to factorization systems","Gluing recollements of extriangulated categories with s-torsion","Bijection unifies s-torsion and factorization in extriangulated"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Extriangulated bijection: s-torsion pairs ↔ factorization systems","New bijection links s-torsion pairs to factorization systems","Gluing recollements of extriangulated categories with s-torsion","Bijection unifies s-torsion and factorization in extriangulated"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000222,"raw_usage":{"total_tokens":1394,"prompt_tokens":823,"completion_tokens":571,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":490}},"tokens_in":439,"tokens_out":571,"duration_ms":6273,"temperature":1.0,"reasoning_tokens":490,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:52:40.112952+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Adachi, H","cited_arxiv_id":null,"evidence_quote":"supplies the definition of s-torsion pairs with negative first extensions and the proposition the proof uses to conclude cone(h) lies in T."},{"cited_title":"Loregian, S","cited_arxiv_id":null,"evidence_quote":"introduces triangulated factorization systems and the bijection with t-structures that this paper generalises."},{"cited_title":"Nakaoka, Y","cited_arxiv_id":null,"evidence_quote":"introduces extriangulated categories, the ambient structure carrying the whole argument."},{"cited_title":"Rosick´ y, W","cited_arxiv_id":null,"evidence_quote":"gives the classical bijection between torsion pairs and normal torsion theories that the new bijection unifies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines recollements of extriangulated categories, which the gluing theorems in Section 4 use."}],"review_version":1}