{"id":"fbe4609b-e1c0-464e-b7df-35569d685b26","arxiv_id":"1908.03546","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A pretorsion theory on any category is a pair of full subcategories (T,F) whose intersection generates an ideal of trivial morphisms, and it yields reflective and coreflective decompositions generalizing classical torsion theories.","lead":"This paper defines pretorsion theories in arbitrary categories, replacing the zero object with the intersection of torsion and torsion-free classes and using morphisms that factor through this intersection. The authors prove that torsion-free parts are epireflective and torsion parts are monocoreflective, and they give examples in preordered sets, endomappings of finite sets, and topological spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 3.4 depends on the paper's explicit global-choice assumption; without a choice-free reformulation the epireflectivity claim is conditional.","rationale":"The reader correctly identifies the Axiom of Choice for classes as the weakest load-bearing point in the proof of the central claim. The paper is explicit about using this axiom, and within a foundation that includes global choice the proof of Corollary 3.4 is sound: the chosen sequences make t and f functors, the universal properties are verified from the Z-prekernel/precokernel conditions, and the unit/counit components are epimorphisms/monomorphisms by Proposition 2.2 and Proposition 2.1. Thus there is no internal mathematical error. The concern is that the theorem as stated is conditional on a strong foundational assumption that is not part of Definition 2.6. Since the paper flags the assumption and standard category theory often works with such choices, this does not change the reader's CONDITIONAL verdict; it reinforces that the result should be understood as proven under the stated foundational convention. The original conditional verdict also rests on novelty and deferred example verifications, which are independent of this concern.","tokens_in":22205,"tokens_out":23419,"duration_ms":266491,"concrete_test":"Re-prove Proposition 3.3(a) without forming the global functor f: for each object X, use the Z-preexact sequence supplied by Axiom (2) and show directly that the map eta_X: X -> f(X) has the universal reflection property with respect to F, and similarly that epsilon_X: t(X) -> X has the coreflection property. Then check whether these objectwise data can be assembled into an adjunction in a choice-free foundation, e.g., by representing t and f as anafunctors. If the objectwise proof succeeds without simultaneous choice, the AC concern is only presentational; if it fails, amend Corollary 3.4 and Propositions 3.3, 3.6, 3.9 and 3.10 to state explicitly that they assume the Axiom of Choice for classes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Corollary 3.4 rests on Proposition 3.3, where for every object X the authors fix a short Z-preexact sequence t(X) -> X -> f(X). Proposition 3.1 shows such sequences are unique only up to non-canonical isomorphism, so choosing one sequence for every object of C is a proper-class choice. The paper acknowledges this and invokes the Axiom of Choice for classes. If that axiom is not accepted, the object maps of the functors t and f are not defined, and the reflector/coreflector conclusion of Corollary 3.4 is not established by the given proof. This is not an internal inconsistency, but it is a genuine foundational dependency: the central structural claim is stated for an arbitrary pretorsion theory, yet its proof requires a strong choice principle not mentioned in Definition 2.6. A direct objectwise universal-property argument might avoid forming global functors, but the paper does not supply one.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces pretorsion theories in an arbitrary category C as a pair (T, F) of full replete subcategories with Z = T ∩ F, subject to two axioms: every morphism from an object of T to an object of F factors through Z, and every object X admits a short Z-preexact sequence A → X → B with A ∈ T and B ∈ F. The paper develops the elementary calculus of Z-prekernels and Z-precokernels, proves that F is epireflective and T is monocoreflective, establishes closure properties under retracts and Z-extensions, analyzes the two idempotent functors ET and EF, gives an abstract characterization of torsion-free subcategories in Proposition 4.5, studies projective objects, and works out examples in preordered sets, finite endomappings and their infinite generalization, finite chains, topological groups, and topological spaces. The main structural theorem is Corollary 3.4, which recovers the two central features of classical torsion theories without assuming that the category is pointed.","tokens_in":22387,"tokens_out":21029,"duration_ms":226422,"significance":"If the results hold, this is a useful and natural unification: classical torsion theories in abelian and homological categories and the ideal-based theories of Grandis and Janelidze appear as special cases, while the framework also covers new examples such as preordered sets, the category of endomappings, and Kolmogorov quotients of topological spaces. The core proofs in Sections 3 through 5 are coherent, and the main theorem—epireflectivity of F and monocoreflectivity of T—is the right categorical generalization. The paper is also honest about its substantial overlap with [18,19]. Its main caveat is a metatheoretic one: the proof of the central adjunction theorems invokes the Axiom of Choice for classes, which is a genuine foundational dependency that should be stated explicitly in the main statements.","major_comments":[{"comment":"The proof that F is epireflective and T is monocoreflective fixes, for every object X of C, a short Z-preexact sequence t(X) → X → f(X) “by the Axiom of Choice for classes.” This global choice is what defines the object maps of t and f and hence the adjunctions in Proposition 3.3; without it, Corollary 3.4 is not a consequence of Definition 2.6 alone. Since Corollary 3.4 is the central structural conclusion, the paper should either state the global-choice assumption explicitly as a hypothesis of Corollary 3.4 and of Propositions 3.6, 3.9, and 3.10, or replace the global choice by a direct objectwise proof of the universal property of each component η_X and ε_X.","section":"Section 3 (before Proposition 3.3) and Corollary 3.4"}],"minor_comments":[{"comment":"The step “βγη_T = η_T = 1_{f(T)}η_T, and βγ = 1_{f(T)}” needs justification: η_T is not proved to be an epimorphism in this lemma. The intended argument is that both 1_{f(T)} and βγ are factorizations of the same morphism through the unit of the reflection, so they coincide by the uniqueness part of the universal property; please say this explicitly.","section":"Lemma 3.5"},{"comment":"After obtaining t = z ∈ Z, the sentence “From Lemma 2.4(a), we get p = f” is not immediate, because Lemma 2.4(a) applies to a morphism that is known to be a Z-precokernel, and the text has not shown that p → f is a Z-precokernel of t → p. Please spell out which Z-preexact sequence is being used (presumably the one supplied by Axiom (2) for the object p) and how t is identified with its torsion part.","section":"Proposition 6.1, proof of (a) ⇒ (b)"},{"comment":"The assertion that (C′, F′) is a pretorsion theory in M′ is stated without proof (“We don’t go too much into details now”); since this example is advertised as a generalization of 6.2, the proof of Axioms (1) and (2), or a precise reference to the analogous proof in [12], should be included or the claim should be marked as an outline.","section":"Section 6.3"},{"comment":"The phrase “Assume that Z is closed under coproducts” should be clarified to mean that the relevant coproducts of families of objects of Z exist and belong to Z; the proof uses existence of a coproduct of the family {Z_j}, not merely closure under existing coproducts.","section":"Proposition 4.3"},{"comment":"There are a few typographical slips: in Section 6.3, “Equivalently, C′ consists of all objects...” should refer to F′; in the diagram in the proof of Corollary 5.3 the top-left object appears to be t(T) rather than T; and the notation “C′” is used inconsistently for the torsion class in Section 6.3. Please correct these.","section":"Typos and notation"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the global-choice issue is real but local. The paper’s own text mentions the Axiom of Choice for classes, so this is a matter of stating the metatheoretic assumption in the statements of the main theorems rather than a mathematical error in the proofs. The paper overlaps substantially with [18,19], but the introduction is candid about this and the examples are sufficiently different. I recommend major revision to address the foundational qualification and the proof clarifications."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the two-line version before anything else. This is a useful framework paper with clean proofs and some genuinely new examples, but the headline novelty is smaller than the title promises once you subtract their own [10] and the Grandis–Janelidze papers, and one central structural claim depends on an explicitly invoked global-choice axiom.\n\nWhat is actually new: the projective-object results in Section 5, the complete classification of pretorsion theories on finite chains (Example 6.4), and the examples around endomappings (finite and infinite), the Kolmogorov quotient, and totally disconnected spaces. The framework itself—pair of replete subcategories T and F, Z = T∩F, ideals of Z-trivial morphisms, Z-prekernels/-precokernels—is a natural generalization, and Sections 3–5 hang together: Proposition 3.3 gives the expected adjunctions, Corollary 3.4 recovers epireflectivity/monocoreflectivity, and the closure properties in Section 4 are correct. The authors are also properly upfront about the overlap with [18] and [19], which counts in their favor.\n\nSoft spots, in proportion. First, the overlap is real, not just formal: the core definitions and several basic results come from their own [10], and the paper would be materially better with an itemized novelty statement. Second, the proofs of some examples are deferred to 'easy to check' or to prior papers; most readers will be fine, but a referee should ask for the finite-chain verification in full, since it is the most substantial new example. Third, the global choice issue: Corollary 3.4 depends on choosing, for every object X, a short Z-preexact sequence t(X)→X→f(X). The authors invoke the Axiom of Choice for classes, and Proposition 3.1 shows uniqueness only up to non-canonical isomorphism. Without that choice, the functors t and f are not defined and the reflector conclusion isn't established by the given proof. This is not an internal inconsistency, but the stated theorem is stronger than the proof supports in a choice-free foundation. A direct objectwise universal-property argument might sidestep it; the paper doesn't supply one.\n\nWho should read it: anyone working on torsion theories in non-abelian categories, radical theory, or closure operators. They'll find the examples useful and the framework convenient.\n\nI would send this to peer review. It deserves refereeing; the main request would be an honest novelty section and either a choice-free proof of Corollary 3.4 or a clear statement that the result is choice-dependent.","headline":"A clean, honest framework paper whose genuinely new content is modest; the main reflector theorem carries a global-choice dependency that the authors acknowledge but don't resolve.","tokens_in":22905,"tokens_out":2351,"would_cite":true,"duration_ms":23075,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18E40","18A40","18B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper defines pretorsion theories in arbitrary categories and proves that every such theory makes the torsion-free part epireflective and the torsion part monocoreflective.","keywords":["pretorsion theory","torsion theory","ideal of morphisms","Z-prekernel","epireflective subcategory","monocoreflective subcategory","preordered sets","endomappings"],"falsifier":"Take a small finite category, such as a finite poset viewed as a category, enumerate all full replete pairs $(\\mathcal T,\\mathcal F)$ satisfying the two pretorsion axioms, and check whether the map from any object $X$ to its chosen $f(X)$ satisfies the universal property of a reflection into $\\mathcal F$. A pair satisfying the axioms but lacking this universal property, or a pair for which the mediating morphisms are not unique, would show that the axioms do not force the epireflective conclusion.","tokens_in":22016,"feed_emoji":"🌀","tokens_out":9820,"duration_ms":99983,"temperature":0.7,"pith_summary":"Torsion theory, classically tied to abelian categories with a zero object, can be rebuilt in any category from a pair of full subcategories $\\mathcal T$ and $\\mathcal F$. The paper calls such a pair a pretorsion theory when every morphism from a torsion object to a torsion-free object factors through the intersection $\\mathcal Z=\\mathcal T\\cap\\mathcal F$, and every object fits into a short $\\mathcal Z$-preexact sequence $t(X)\\to X\\to f(X)$ with $t(X)\\in\\mathcal T$, $f(X)\\in\\mathcal F$. The central result is that these minimal axioms force the torsion-free part to be epireflective and the torsion part to be monocoreflective, recovering the structural core of classical torsion theories without kernels, cokernels, or a zero object. The paper also documents genuinely new examples in finite endomappings, preordered sets, topological groups, and topological spaces, showing the framework is broader than the pointed or homological settings.","feed_headline":"Torsion theories no longer need a zero object","feed_subtitle":"Reframing torsion with any pair of subcategories gives reflectors without kernels or cokernels.","key_machinery":"The machinery is the ideal of $\\mathcal Z$-trivial morphisms—maps that factor through an object of $\\mathcal Z$—together with the relative notions of $\\mathcal Z$-prekernel, $\\mathcal Z$-precokernel, and short $\\mathcal Z$-preexact sequence. A short $\\mathcal Z$-preexact sequence $A\\to B\\to C$ is a pair where the first arrow is the $\\mathcal Z$-prekernel of the second and the second is the $\\mathcal Z$-precokernel of the first; these replace ordinary kernels and cokernels. The universal properties of these prekernels and precokernels are what make the arrows $t(\\phi)$ and $f(\\phi)$ exist and be functorial, and that functoriality is what produces the adjunctions behind the reflection and coreflection.","core_discovery":"On the paper's own terms, the discovery is that the whole torsion-theory machine can be run with no zero object. Given any pretorsion theory $(\\mathcal T,\\mathcal F)$ in a category $\\mathcal C$ with $\\mathcal Z=\\mathcal T\\cap\\mathcal F$, the torsion-free subcategory $\\mathcal F$ is epireflective and the torsion subcategory $\\mathcal T$ is monocoreflective; equivalently, the inclusions $\\mathcal F\\to\\mathcal C$ and $\\mathcal T\\to\\mathcal C$ have adjoints whose units and counits are epimorphisms and monomorphisms. The canonical sequence $t(X)\\to X\\to f(X)$ is unique up to unique isomorphism, $t$ and $f$ are idempotent functors, and the analogue of the classical identity $t(A/t(A))=0$ holds in the form that both $f(t(X))$ and $t(f(X))$ lie in $\\mathcal Z$ for every $X$.","pith_inferences":["This suggests that the pretorsion sequence in the endomapping category packages the usual decomposition of a functional graph into cycles (the torsion part) and a forest (the torsion-free part); the paper does not spell out this normal-form reading, but it follows from the functoriality of $t$.","Because the comparison morphism $f(t(X))\\to t(f(X))$ can fail to be an isomorphism, the size of its failure is a plausible measure of how far a given pretorsion theory is from being abelian-like; classifying the theories where it is an isomorphism is a natural open question.","For concrete categories, Proposition 4.5 can be read as a certification recipe: to exhibit a pretorsion theory it is enough to name a $\\mathcal Z$-normal epireflective subcategory whose unit has prekernels and whose induced functor is idempotent, so the framework reduces the search to those three checks."],"forward_implications":["Every pretorsion theory gives an idempotent torsion functor $t$ and torsion-free functor $f$, so each object decomposes functorially into a torsion part and a torsion-free quotient.","The torsion-free part $\\mathcal F$ is closed under all limits and the torsion part $\\mathcal T$ under all colimits, matching the classical closure properties in abelian settings.","The three classes $\\mathcal T$, $\\mathcal F$, and $\\mathcal Z$ are closed under retracts and under $\\mathcal Z$-extensions, so the decomposition is preserved by the basic ways of building new objects from old.","An object is torsion if and only if it is isomorphic to $t(X)$ for some $X$, and torsion-free if and only if it is isomorphic to $f(X)$; the two functors therefore classify the two halves of the category.","When $\\mathcal Z$ is the class of projective objects, the pretorsion axioms reduce to checking that $\\mathcal T$ is closed under extremal quotients and $\\mathcal F$ under subobjects, giving a concrete criterion for many categories."],"supporting_citations":[{"why":"Sets up the definitions of $\\mathcal Z$-trivial morphisms, $\\mathcal Z$-prekernels, $\\mathcal Z$-precokernels, and short $\\mathcal Z$-preexact sequences, which the paper's axioms reuse as primitive material.","marker":"[10]"},{"why":"Supplies the classical homological-category torsion theory with a zero object that the new axioms generalize, together with topological-group examples used for comparison.","marker":"[2]"},{"why":"Provides the closest prior ideal-based torsion theory in multi-pointed categories, against which the paper positions its weaker hypotheses.","marker":"[18]"},{"why":"Gives the pointed-category torsion theory whose normal-epireflective characterization is shown in the paper to be a special case of Proposition 4.5.","marker":"[21]"},{"why":"Provides the finite endomapping example, whose graph-theoretic structure is reinterpreted as a pretorsion theory and then extended to arbitrary sets.","marker":"[12]"},{"why":"Defines ideals of morphisms, the notion underlying the paper's $\\mathcal Z$-trivial morphisms and their ideal property.","marker":"[8]"}],"fun_headline_variants":["Torsion theories go zero-free","Pretorsion: torsion without a zero object","General torsion: no zero needed","Ditching the zero object for torsion","Zero-free torsion for all categories"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the required short $\\mathcal Z$-preexact sequence $t(X)\\to X\\to f(X)$ can be chosen simultaneously for every object $X$; the paper invokes the Axiom of Choice for classes to get the functors $t$ and $f$, and without that simultaneous choice the reflector and coreflector conclusions are not established.","fun_headline_variants_meta":{"raw":{"variants":["Torsion theories go zero-free","Pretorsion: torsion without a zero object","General torsion: no zero needed","Ditching the zero object for torsion","Zero-free torsion for all categories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000304,"raw_usage":{"total_tokens":1740,"prompt_tokens":935,"completion_tokens":805,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":744}},"tokens_in":551,"tokens_out":805,"duration_ms":7374,"temperature":1.0,"reasoning_tokens":744,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:11:05.003125+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small finite category, such as a finite poset viewed as a category, enumerate all full replete pairs $(\\mathcal T,\\mathcal F)$ satisfying the two pretorsion axioms, and check whether the map from any object $X$ to its chosen $f(X)$ satisfies the universal property of a reflection into $\\mathcal F$. A pair satisfying the axioms but lacking this universal property, or a pair for which the mediating morphisms are not unique, would show that the axioms do not force the epireflective conclusion.","supporting_citations":[{"cited_title":"Facchini and C","cited_arxiv_id":null,"evidence_quote":"Sets up the definitions of $\\mathcal Z$-trivial morphisms, $\\mathcal Z$-prekernels, $\\mathcal Z$-precokernels, and short $\\mathcal Z$-preexact sequences, which the paper's axioms reuse as primitive material."},{"cited_title":"Bourn and M","cited_arxiv_id":null,"evidence_quote":"Supplies the classical homological-category torsion theory with a zero object that the new axioms generalize, together with topological-group examples used for comparison."},{"cited_title":"Grandis and G","cited_arxiv_id":null,"evidence_quote":"Provides the closest prior ideal-based torsion theory in multi-pointed categories, against which the paper positions its weaker hypotheses."},{"cited_title":"Categories in algebra, geometry and mathematical physics","cited_arxiv_id":null,"evidence_quote":"Gives the pointed-category torsion theory whose normal-epireflective characterization is shown in the paper to be a special case of Proposition 4.5."},{"cited_title":"Facchini and L","cited_arxiv_id":null,"evidence_quote":"Provides the finite endomapping example, whose graph-theoretic structure is reinterpreted as a pretorsion theory and then extended to arbitrary sets."},{"cited_title":"Oeuvres comp l` etes et comment´ ees","cited_arxiv_id":null,"evidence_quote":"Defines ideals of morphisms, the notion underlying the paper's $\\mathcal Z$-trivial morphisms and their ideal property."}],"review_version":1}