Pretorsion theories in general categories

Alberto Facchini; Carmelo Finocchiaro; Marino Gran

REVIEW 1 major objections 5 minor 30 references

Pretorsion theories in general categories

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.03546 v2 pith:WMT7FRRF submitted 2019-08-09 math.CT math.ACmath.RA

Alberto Facchini , Carmelo Finocchiaro , Marino Gran This is my paper · ORCID

classification math.CTmath.ACmath.RA MSC 18E4018A4018B35

keywords pretorsiontheorytorsionidealofmorphismsZ-prekernelepireflectivesubcategorymonocoreflectivepreorderedsetsendomappings

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $X$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

What 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.

Soft 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.

Who 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.

I 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.

Referee Report

1 major / 5 minor

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.

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 (1)

[Section 3 (before Proposition 3.3) and Corollary 3.4] 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.

minor comments (5)

[Lemma 3.5] 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.
[Proposition 6.1, proof of (a) ⇒ (b)] 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 6.3] 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.
[Proposition 4.3] 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.
[Typos and notation] 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.

Circularity Check

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No circularity: Corollary 3.4 is a direct consequence of Definition 2.6; self-citations concern only elementary parameter-free lemmas.

full rationale

The central claim Corollary 3.4 follows from the axioms of a pretorsion theory without any fitted data or imported uniqueness. By Definition 2.6(2), every object B admits a short Z-preexact sequence t(B) -> B -> f(B) with f(B) in F. Since t(B) is in T and any F in F satisfies hom(T,F) = Triv_Z(T,F) by Definition 2.6(1), the composite t(B) -> B -> F is Z-trivial; the universal property of the Z-precokernel eta_B then gives the unique factorization through f(B), so f is a reflection. Proposition 2.2 supplies that eta_B is an epimorphism, giving epireflectivity. The use of the Axiom of Choice for classes to choose the sequences is explicitly stated in Section 3 and is a foundational condition on the proof, not a circular reduction. The paper's citations of the authors' own [10] for Propositions 2.1, 2.2, 2.7 and Lemma 2.4 concern parameter-free lemmas about Z-prekernels and orthogonality whose assumptions do not include the target reflectivity result; under the review rules these do not raise the circularity score. The acknowledged overlap with [18,19] is an external comparison, not the basis of the derivation. No step was found in which a prediction or first-principles result reduces by construction to its inputs.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The framework rests on the two axioms of Definition 2.6 and on the Axiom of Choice for classes used to define the reflections.

assumptions (3)

standard math Axiom of Choice for classes
Invoked in Section 3 before Proposition 3.3 to select a short Z-preexact sequence for each object X; standard in category theory for class-sized categories.
domain assumption Definition 2.6(1): hom_C(T,F) = Triv_Z(T,F) for T in T, F in F
Postulates that every morphism from a torsion object to a torsion-free object factors through Z = T intersect F; this is the orthogonality condition of the framework.
domain assumption Definition 2.6(2): every object B admits a short Z-preexact sequence A -> B -> C with A in T and C in F
Postulates existence of the decomposition; all the adjunction and closure results in Sections 3 and 4 depend on it.

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Pith. "Pith review of Pretorsion theories in general categories." pith.science (2026). https://pith.science/paper/WMT7FRRF

@misc{pith2026190803546,
  author       = {Pith},
  title        = {Pith review of: Pretorsion theories in general categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMT7FRRF}},
  note         = {Machine review of arXiv:1908.03546}
}

abstract

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair ($\mathcal T$, $\mathcal F$) of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = \mathcal T \cap \mathcal F$ of \emph{trivial objects} in $\mathcal C$. The morphisms which factor through $\mathcal Z$ are called $\mathcal Z$-trivial, and these form an ideal of morphisms, with respect to which one can define $\mathcal Z$-prekernels, $\mathcal Z$-precokernels, and short $\mathcal Z$-preexact sequences. This naturally leads to the notion of pretorsion theory, which is the object of study of this article, and includes the classical one in the abelian context when $\mathcal Z$ is reduced to the $0$-object of $\mathcal C$. We study the basic properties of pretorsion theories, and examine some new examples in the category of all endomappings of finite sets and in the category of preordered sets.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

Facchini and C

A. Facchini and C. Finocchiaro, Pretorsion theories, stable category and preordered sets, Ann. Mat. Pura Appl. 199 (2020), 1073–1089, https://doi.org/10.1007/s10231-019 -00912-2

work page doi:10.1007/s10231-019 2020

[2] [18]

Grandis and G

M. Grandis and G. Janelidze, From torsion theories to closure operators and factorizati on systems, Categories and General Algebraic Structures with Applicat ions, 12(1) (2020), 89– 121

work page 2020

[3] [19]

Grandis, G

M. Grandis, G. Janelidze and L. M´ arki, Non-pointed exactness, radicals, closure operators, J. Aust. Math. Soc. 94(3) (2013), 348–361, doi:10.1017/S1446788713000086

work page doi:10.1017/s1446788713000086 2013

[4] [1]

Barr, Non-abelian torsion theories, Canad

M. Barr, Non-abelian torsion theories, Canad. J. Math. 25 (1973), 1224–1237, https://doi.org/10.4153/CJM-1973-130-4

work page doi:10.4153/cjm-1973-130-4 1973

[5] [2]

Bourn and M

D. Bourn and M. Gran, Torsion theories in homological categories , J. Algebra 305 (2006), 18–47, https://doi.org/10.1016/j.jalgebra.2006.07.01 1

work page doi:10.1016/j.jalgebra.2006.07.01 2006

[6] [3]

A. Buys, N. J. Groenewald and S. Veldsman, Radical and semisim- ple classes in categories, Quaestiones Math. 4(3) (1980–81), 205–220, https://doi.org/10.1080/16073606.1981.9631873

work page arXiv 1980

[7] [4]

Buys and S

A. Buys and S. Veldsman, Quasiradicals and radicals in categories, Publ. Inst. Math. (Beograd) (N.S.) 38(52) (1985), 51–63

work page 1985

[8] [5]

M. M. Clementino, D. Dikranjan and W. Tholen, Torsion theories and radicals in normal categories, J. Algebra 305 (2006), 92–129, https://doi.org/10.1016/j.jalgebra.20 05.09.030

work page doi:10.1016/j.jalgebra.20 2006