Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories

Federico Campanini; Laura Cossu

Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories

T0 review · 0 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The category of atomic monoids is locally finitely presentable, and its compact objects are exactly finitely presented monoids.

desk verdict Solid paper: AtoMon is locally finitely presentable and not regular, with explicit proofs that hold up; send it to peer review. read the letter →

arxiv 2607.23144 v1 pith:QOV7AKHR submitted 2026-07-25 math.CT math.RA

Federico Campanini , Laura Cossu This is my paper · ORCID

classification math.CTmath.RA MSC 18A0518A3218A4018C3518E0818E40

keywords atomicmonoidlocallypresentablecategoryregularstronggeneratorcompactobjectadjunctionpretorsiontheoryfactorization

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

This paper establishes that AtoMon, the category of atomic monoids and atom-preserving homomorphisms, is locally finitely presentable even though it is not regular and not a variety of universal algebras. It exhibits a small strong generator made of compact objects—free monoids on one generator, the infinite cyclic group, and single-relation atomic monoids M(X;ω1,ω2)—and shows that the compact objects of AtoMon are precisely the finitely presented atomic monoids. The paper also constructs a canonical (regular epi, mono)-factorization, proves that regular epimorphisms are not pullback-stable via an explicit counterexample, and builds adjunctions for the group-of-units functor and an atomization functor. Finally, it lifts torsion theories of groups to pretorsion theories of atomic monoids, yielding the example (Grp, RedAtoMon), and gives a general criterion for such lifts along monocoreflections. A sympathetic reader should care because these results place the central objects of factorization theory inside a finitely presentable categorical framework with explicit universal constructions, while precisely locating where the category departs from algebraic variety behavior.

What carries the argument

The load-bearing machinery is the strong generator G defined in equation (1): the union of {F({x}), G} (free monoid on one generator and infinite cyclic group) with all monoids M(X;ω1,ω2) presenting a single relation between two distinct words of length at least two. These M(X;ω1,ω2) act as 'relation detectors': they are reduced atomic monoids whose atoms are exactly the generators, and a morphism from such an object into an atomic monoid K encodes an equality between two products of atoms of K. This lets the strong-generator proof force any candidate monomorphism to be injective on reducible non-units, not just on units and atoms. The paper also uses the characterization, imported from its

What would settle it

One direct test: in the pullback construction of Section 3.2, compute the quotient P/θ_ρ. The paper asserts that the pair (A1B, C3D) is identified by the ordinary monoid kernel of ρ but not by the categorical kernel congruence θ_ρ; a direct computation—or finding such a pair that is in θ_ρ—would settle whether ρ is a regular epimorphism. Independently, exhibiting an atom-preserving homomorphism that is injective on H^× ∪ A(H) but not a monomorphism in AtoMon would invalidate the monomorphism characterization used in both main proofs.

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Extended reading notes

Core claim

The central claim is that AtoMon is locally finitely presentable, proved by exhibiting a strong generator G consisting entirely of compact objects: the free monoid on one generator, the infinite cyclic group, and monoids M(X;ω1,ω2) = ⟨X | ω1=ω2⟩ where ω1 and ω2 are distinct words of length at least two over a finite set X. Because the generator is strong and all its objects are compact, the standard criterion for local finite presentability applies. The paper further shows that compact objects in AtoMon are exactly atomic monoids that are finitely presented as ordinary monoids. On regularity, it proves that every morphism admits a (regular epi, mono)-factorization, but constructs an explicit

Load-bearing premise

The proofs of local presentability and non-regularity both rely on the imported characterization that a morphism is a monomorphism in AtoMon exactly when it is injective on units and atoms; if that characterization fails, the strong-generator proof and the pullback counterexample collapse.

Editorial extensions

If this is right

If the central claims are correct, AtoMon is equivalent to the category of models of a finitary finite-limit theory, hence a finitary essentially algebraic category, even though it is not a variety.
Every atomic monoid arises as a filtered colimit of compact objects from the closure of G under finite colimits, giving a canonical presentation-theoretic handle on arbitrary atomic monoids.
Compact objects in AtoMon coincide with finitely presented monoids, so categorical finiteness matches ordinary monoid finite presentability.
The explicit non-regular epimorphism shows that the regular completion of AtoMon contains a genuinely new object, not isomorphic to any atomic monoid, which must be adjoined to make pullback-stable factorizations possible.
Every torsion theory of groups induces a pretorsion theory of atomic monoids, so group-theoretic torsion phenomena have a direct analogue in the arithmetic of atomic monoids.

Reading between the lines

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

The strong generator's single-relation monoids suggest that equalities between products of atoms—the core of factorization theory—are finitely axiomatizable, so arithmetic invariants of atomic monoids may be studied through finitary finite-limit logic.
The failure of regularity is intimately tied to the fact that ordinary monoid congruences can produce quotients that are not atomic (as in the paper's Example 3.1); the categorical kernel pair in AtoMon is designed to avoid this, but at the cost of not being an ordinary equivalence relation on the underlying set.
The pretorsion-lifting criterion is likely applicable beyond groups and monoids: any monocoreflective subcategory whose torsionfree objects are detected by the right adjoint should yield a pretorsion theory, so one can test the construction on other algebraic structures with a 'units-like' invariant.

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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

Short version: this is a good paper. The headline result—AtoMon is locally finitely presentable—is proved with a genuinely constructed strong generator, and the counterexample to regularity is explicit and checkable. The compact-object theorem and the two adjunctions are clean. The pretorsion-lifting section is a nice bonus.

What is actually new: the strong generator G in (1) is original and does real work; the pullback counterexample in Prop 3.7 is the first explicit failure of regularity, and the exponent-vector computation in Lemma 3.6 is a sound way to prove injectivity. The adjunctions for the units functor and the atomization functor are natural, and the general lifting criterion in Theorem 5.6 is a useful abstraction beyond the monoid setting.

Soft spots, in proportion: the paper leans heavily on the authors' own [3] for two facts—colimits in AtoMon are computed as in Mon, and a monomorphism is exactly a map injective on units and atoms. That dependency is clearly cited, and the monomorphism characterization can be verified directly from the definitions, so I do not treat it as a load-bearing flaw. Still, a referee should check those imports because the strong-generator proof and the non-regularity example rest on them. Minor: the atomicity of H/θ_H in Prop 5.3 is asserted without proof; it is easy, but should be spelled out. The notation clash for '1' (identity element vs terminal object) is a clarity issue. The regular completion subsection is more descriptive than deep—fine, but not the core of the paper.

On the citation pattern: the self-citations are to earlier independent work, not an attempt to recycle conclusions. No circularity, no data fitting. The central proofs are coherent and detailed; the reader's check and the stress test both hold up, and I agree with them.

Who this is for: people working on categorical factorization theory, categories of monoids, or pretorsion theories. It deserves a serious referee. I would send it to peer review without hesitation.

Referee Report

0 major / 5 minor

Summary. The paper studies the category AtoMon of atomic monoids and atom-preserving homomorphisms. It proves that AtoMon is locally finitely presentable by explicitly constructing a strong generator G (Eq. 1) consisting of compact objects, and it identifies the compact objects of AtoMon with finitely presented atomic monoids (Thm 2.7). It then shows that AtoMon admits (regular epi, mono)-factorizations (Prop 3.3) but is not a regular category: an explicit regular epimorphism p whose pullback along a monomorphism is not regular is constructed and verified in detail (Props 3.5--3.7). The paper also establishes adjunctions involving the group-of-units functor (Prop 4.1), constructs an explicit right adjoint to the underlying-monoid functor (Prop 4.2), and lifts torsion theories of Grp to pretorsion theories of AtoMon (Prop 5.3), with a general lifting criterion for monocoreflective subcategories (Thm 5.6).

Significance. The results, if correct, give a clean structural picture of AtoMon: a locally finitely presentable category that nevertheless fails regularity. The explicit strong generator is a useful and nontrivial tool, and the non-regularity counterexample is concrete and checkable, with exponent-vector computations that I verified. The adjunction and pretorsion results broaden the categorical toolkit for factorization theory. I checked the load-bearing arguments—strong generator proof, compact-objects theorem, non-regularity counterexample, and adjunctions—and found them sound. The paper is careful in citing the necessary infrastructure from prior work [3], and it provides explicit constructions rather than mere existence arguments.

minor comments (5)

[Prop 5.3] The proof asserts without proof that H/θ_H is atomic and that ρ_H is a morphism in AtoMon, and later uses (H/θ_H)^× ≅ H^×/t(H^×). These facts are needed for the pretorsion construction. A short argument showing that the congruence only identifies unit multiples of atoms (so quotient atomicity and atom-reflection follow) would make the section self-contained.
[Lemma 3.6] The map Φ:M→P is shown to be a bijective monoid homomorphism, but to conclude it is an isomorphism in AtoMon one should also note that Φ preserves atoms. This is true because the generators of M map to the generating pairs (3), which are atoms of P; adding this one-line check would remove an implicit step.
[Prop 2.4] The assertion that {F,G} is a generator is stated without proof. It is easy: if two morphisms differ on an element, they differ on some atom or unit in a factorization of that element. A brief justification would improve readability.
[§4.2 and Remark 4.3] The letter A is used both for the new atomization functor A:Mon→AtoMon and for the atom-set functor A:AtoMon→Set. This overloading is confusing in Remark 4.3. Consider using different notation, e.g. 𝔸 for the atomization functor.
[General] There is a duplicated word in §5.2: 'we apply the criterion of Theorem 5.6 to a a context different' should read 'to a context different'. Also, the regular completion subsection would benefit from a sentence explaining why the equivalence relation on representatives is well-defined under composition; this follows from the defining condition gα=gα', but it is not spelled out.

Circularity Check

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No significant circularity: the derivation is self-contained modulo prior independent results cited from [3].

full rationale

The paper's derivation chain is not circular. Theorem 2.6 is proved from the standard strong-generator criterion ([1, Theorem 1.11]) by constructing the set G in (1) and proving in Proposition 2.4 that it is a strong generator. That proof is an honest construction: it uses the monomorphism criterion from [3, Remark 3.1(iv)] and the colimit computation from [3, Theorem 6.3] as prior lemmas about AtoMon, not as instances of the LFP conclusion being proved. The non-regularity counterexample (Proposition 3.7) computes the pullback by [3, Proposition 6.5] and then verifies by explicit exponent-vector arithmetic that (x,y) is not in θ_ρ; this is a concrete computation, not a restatement of an assumption. The adjunction and pretorsion sections construct objects (A(M), I(G), H/θ_H) and verify the required universal properties directly; Theorem 5.6 is a general criterion that is then applied to AtoMon in Remark 5.7, not a conclusion presupposed by the construction. The self-citations to [3] are to earlier independent results with stated assumptions; under the review rules, such citations are real evidence and do not raise the circularity score. No fitted value is renamed as a prediction, no equation is equal to its input by construction, and no uniqueness theorem from the authors is used to forbid alternatives.

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

The central claims rest on standard categorical theorems and on the authors' prior paper [3] for the concrete structure of AtoMon (colimits, monomorphisms, kernel pairs). These are external premises, clearly cited, and not invented here. No free parameters or fitted values appear; the paper is purely structural mathematics.

assumptions (6)

domain assumption Every atomic monoid is Dedekind-finite: a product is a unit iff each factor is a unit ([8, Prop 2.30]).
Used in Section 1.1 and in the adjunction proofs (Prop 4.1, 4.2) to classify products of units, atoms, and reducibles.
domain assumption Colimits in AtoMon are computed as in the category of monoids; limits generally are not ([3, Theorems 6.3, 6.6]).
Used in Lemma 2.2 and Theorem 2.7 to transfer compactness from Mon to AtoMon.
domain assumption A morphism in AtoMon is a monomorphism iff it is injective on H^× ∪ A(H) ([3, Remark 3.1(iv)]).
Used in Proposition 2.4 and Lemma 3.5; this is the weakest load-bearing external premise.
domain assumption The kernel pair of a morphism in AtoMon is the monoid generated by pairs of atoms or units with equal images ([3, Theorem 6.5]).
Used in Section 3 to define θ_f and characterize regular epimorphisms.
standard math A category is locally finitely presentable iff it is cocomplete and has a strong generator consisting of compact objects ([1, Theorem 1.11]).
Used in Theorem 2.6.
standard math Finite colimits of compact objects are compact in a locally finitely presentable category ([1, Proposition 1.3]).
Used in Theorem 2.7.

invented entities (2)

Atomization functor A: Mon → AtoMon independent evidence
purpose: Right adjoint to the underlying monoid functor U; explicitly constructs the cofree atomic monoid over a monoid M as ({1}×M^×) ∪ ({a,0}×M).
Fully constructed and the adjunction U ⊣ A is proved in Proposition 4.2; not a postulated entity without evidence.
Trivial ideal extension functor I: Grp → AtoMon independent evidence
purpose: Right adjoint to the group-of-units functor (-)^×; sends a group G to G ⊘ T where T={0,a|a^2=0} with G acting trivially on T.
Explicitly defined in Section 4.1 and the adjunction (-)^× ⊣ I is proved in Proposition 4.1.

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Pith. "Pith review of Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories." pith.science (2026). https://pith.science/paper/QOV7AKHR

@misc{pith2026260723144,
  author       = {Pith},
  title        = {Pith review of: Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QOV7AKHR}},
  note         = {Machine review of arXiv:2607.23144}
}

abstract

We study the category $\mathsf{AtoMon}$ of atomic monoids and atom-preserving homomorphisms. We prove that $\mathsf{AtoMon}$ is locally finitely presentable by exhibiting a strong generator consisting of compact objects. We show that $\mathsf{AtoMon}$ admits (regular epi, mono)-factorizations but that it is not a regular category: we construct a regular epimorphism which is not pullback-stable. We also establish adjunctions for the group of units and explicitly construct the ``cofree atomic monoid'' over an arbitrary monoid. Finally, we exhibit a way to lift torsion theories of $\mathsf{Grp}$ to pretorsion theories of $\mathsf{AtoMon}$ and extend this construction to a more general setting.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 1 linked inside Pith

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [3]

Campanini, L

F. Campanini, L. Cossu, and S. Tringali,The category of atomic monoids: universal constructions and arithmetic properties, arXiv:2502.06610

arXiv

[2] [1]

Adámek and J

J. Adámek and J. Rosický,Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994

1994

[3] [2]

Borceux,Handbook of Categorical Algebra 2: Categories and Structures, Encyclopedia of Mathematics and its Applications, vol

F. Borceux,Handbook of Categorical Algebra 2: Categories and Structures, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, Cambridge, 1994

1994

[4] [4]

Carboni and E

A. Carboni and E. M. Vitale,Regular and exact completions, J. Pure Appl. Algebra125, no. 1-3 (1998), 79–116

1998

[5] [5]

Cossu and S

L. Cossu and S. Tringali,On the arithmetic of power monoids, J. Algebra686(2026), 793–813

2026

[6] [6]

S. E. Dickson,A torsion theory for abelian categories, Trans. Amer. Math. Society121(1966) pp. 223–235

1966

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Facchini, C

A. Facchini, C. A. Finocchiaro and M. Gran,Pretorsion theories in general categories, J. Pure Appl. Algebra225no. 2 (2021) 106503

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[8] [8]

Fan and S

Y. Fan and S. Tringali,Power monoids: a bridge between factorization theory and arithmetic combinatorics, J. Algebra 512(2018), 252–294

2018