{"id":"e338b89d-8e2d-4d6e-8ae8-73565640319e","arxiv_id":"2607.23144","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The category of atomic monoids is locally finitely presentable, non-regular, with explicit adjunctions and a general pretorsion-lifting theorem.","lead":"The paper proves that the category of atomic monoids is locally finitely presentable, not regular, and supports adjunctions for units and cofree atomic monoids. It also gives a general method to lift group torsion theories to pretorsion theories, useful for factorization theory and categorical algebra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader identified the imported monomorphism characterization as the weakest assumption. I agree that this is an external dependency, but a direct proof from the definition of AtoMon shows it is true and easy: if a morphism is injective on units and atoms, it is a monomorphism because all elements are products of units and atoms; conversely, non-injectivity on atoms or units yields distinct maps from F or G. Thus it is not a genuine load-bearing concern. The more substantive dependency is the colimit computation from [3], which underpins compactness and LFP. I checked the main proofs of the paper and found no internal inconsistency. The non-regularity counterexample is valid; the pretorsion lifting, while having a small unproved claim about atomicity, is not central to the paper's headline claims. Therefore, I do not raise a significant objection, and the verdict remains unchanged.","tokens_in":16808,"tokens_out":24767,"duration_ms":219677,"concrete_test":"Independently verify [3, Theorem 6.3] for a nontrivial filtered colimit, e.g., the chain of free monoids F({x1,...,xn}) with the canonical inclusions n→n+1. Compute the colimit in AtoMon using the explicit construction from [3] and compare it with the colimit in Mon (the free monoid on countably many generators). If the two colimits differ, Lemma 2.2 and Theorem 2.6 lose their foundation; if they agree, the central LFP proof is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reviewing the central proofs, I find no load-bearing flaw. The strong generator proof (Prop 2.4) is internally sound: the argument that a non-injective monomorphism would force two distinct atomic factorizations to collapse via the constructed M(X; ω1, ω2) is valid, and the reliance on the monomorphism characterization from [3, Remark 3.1(iv)]—which the reader flagged—can be verified directly from the definition of AtoMon, so it is not a genuine vulnerability. The non-regularity counterexample (Prop 3.7) computes the pullback via [3, Prop 6.5] and checks injectivity with exponent vectors; the square-free relation makes the 'there exists W' step valid, and the distinctness of AB and CD in the quotient is correct. The compact-objects theorem (Thm 2.7) depends on colimits in AtoMon being computed as in Mon ([3, Thm 6.3]); this is the least self-contained point, but it is clearly cited and the surrounding argument is consistent. Minor unproved claims, such as atomicity of H/θ_H in Prop 5.3, are not load-bearing for the main LFP/regularity results and can be proved by a short auxiliary argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":16949,"tokens_out":51198,"duration_ms":494170,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Prop 5.3"},{"comment":"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.","section":"Lemma 3.6"},{"comment":"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.","section":"Prop 2.4"},{"comment":"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.","section":"§4.2 and Remark 4.3"},{"comment":"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.","section":"General"}],"recommendation":"minor_revision","confidential_remarks":"I found no load-bearing errors. The main proofs are sound and the explicit counterexamples are convincing. The small gaps in Section 5 and the notational issues are local and easily fixed. I recommend acceptance after minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSoft 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.\n\nOn 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.\n\nWho 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.","headline":"Solid paper: AtoMon is locally finitely presentable and not regular, with explicit proofs that hold up; send it to peer review.","tokens_in":17558,"tokens_out":1799,"would_cite":true,"duration_ms":19676,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18A05","18A32","18A40","18C35","18E08","18E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The category of atomic monoids is locally finitely presentable, and its compact objects are exactly finitely presented monoids.","keywords":["atomic monoid","locally presentable category","regular category","strong generator","compact object","adjunction","pretorsion theory","factorization theory"],"falsifier":"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.","tokens_in":16546,"feed_emoji":"⚛️","tokens_out":4580,"duration_ms":42206,"temperature":0.7,"pith_summary":"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.","feed_headline":"Atomic monoids form a locally presentable category","feed_subtitle":"A strong generator of compact objects organizes the category, yet regularity still fails.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Atomic monoids: locally presentable, yet not regular","Category of atomic monoids is locally presentable, not regular","Atomic monoids: compact generator but regularity fails","Locally presentable atomic monoids: not regular","Pretorsion theories from groups to atomic monoids"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Atomic monoids: locally presentable, yet not regular","Category of atomic monoids is locally presentable, not regular","Atomic monoids: compact generator but regularity fails","Locally presentable atomic monoids: not regular","Pretorsion theories from groups to atomic monoids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000332,"raw_usage":{"total_tokens":1643,"prompt_tokens":666,"completion_tokens":977,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":899}},"tokens_in":410,"tokens_out":977,"duration_ms":7703,"temperature":1.0,"reasoning_tokens":899,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T03:30:09.102610+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}