{"id":"b319d8e2-6ea8-4efa-b048-70ca34191330","arxiv_id":"2502.06610","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The category of atomic monoids with atom-preserving homomorphisms is complete and cocomplete, with explicit product, coproduct, equalizer, and pullback constructions and length-set formulas.","lead":"This paper defines AtoMon, the category whose objects are atomic monoids and whose arrows are homomorphisms that send atoms to atoms, and proves it has all limits and colimits. It also gives explicit formulas for products, coproducts, and factorization length sets, laying groundwork for realization problems in factorization theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.1's coequalizer proof omits a verification that is load-bearing for the universal property: non-unit non-atoms of K must not become atoms of Q, and this requires a short argument using atomicity and the unit characterization.","rationale":"The reader's weakest-assumption analysis correctly located Proposition 6.1's unproved claim. I agree that it is the least explicit load-bearing step in the cocompleteness proof: without knowing that atoms of Q are exactly the images of atoms of K, the induced morphism in the coequalizer universal property might fail to be atom-preserving. However, the missing argument is short and fully supplied by the existing ingredients: atomicity of K and the unit characterization of Q already proved in the same proposition. Thus the concern is a genuine but minor gap, not a fatal flaw. The remaining constructions—products in Theorem 5.4, equalizers in Proposition 6.4, coproducts in Theorem 4.6, and the use of coequalizers and coproducts for cocompleteness—are consistent and I found no counterexample to the central claim that AtoMon is complete and cocomplete. I also noticed a separate typo in Proposition 6.5: the defining condition for UP is printed as f(x)=g(x), while the proof and the pullback square require f(x)=g(y). This does not affect Theorem 6.6, which relies on products and equalizers, but the proposition should be corrected editorially. None of this changes the reader's ACCEPT verdict.","tokens_in":19944,"tokens_out":27103,"duration_ms":258309,"concrete_test":"Insert the missing argument into Proposition 6.1: for x∈K∖(K×∪A(K)), take an atomic factorization x=a1⋯an with n≥2, set u=a1 and v=a2⋯an, and use the already-established unit characterization to conclude [u] and [v] are non-units of Q. Then verify that the unique induced monoid map ψ:Q→M in the coequalizer universal property sends each atom of Q to an atom of M, using A(Q)={[a]:a∈A(K)} together with φ being atom-preserving. If this one-line verification succeeds, the cocompleteness proof is complete as intended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 6.1 is the load-bearing step for cocompleteness. The proof establishes the unit characterization of Q and shows that atoms of K project to atoms of Q. It then states, without proof, that if x is a non-unit non-atom of K then [x] is not an atom of Q, calling this 'immediate to check'. This claim is genuinely needed, not merely for atomicity of Q, but for the universal property: the induced map ψ:Q→M from the coequalizer must be atom-preserving. If [x] were an atom of Q for some non-atom x, then ψ([x])=φ(x) would be a product of at least two atoms of M and hence not an atom of M, so ψ would fail to be a morphism in AtoMon. The missing verification is short: since K is atomic, x=a1⋯an with n≥2; put u=a1 and v=a2⋯an, both non-units. By the unit characterization, [u] and [v] are non-units of Q, so [x]=[u][v] is a product of two non-units and therefore not an atom. Supplying this line completes the coequalizer proof; no counterexample to the main theorem surfaced.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces AtoMon, the category of atomic monoids with atom-preserving homomorphisms, and studies its categorical structure. The main results are: explicit constructions of products (Theorem 5.4) as the submonoid of the direct product generated by tuples that are either all units or all atoms, and coproducts (Theorem 4.6) as free products; existence of equalizers (Proposition 6.4) and coequalizers (Proposition 6.1), leading to completeness (Theorem 6.6) and cocompleteness (Theorem 6.3); and formulas for length sets and systems of length sets of products and coproducts (Theorems 4.10, 4.11, 5.6, 5.7, 5.8). The paper also establishes an adjunction between the atom functor and the free monoid functor (Proposition 3.4) and shows that acyclicity, unit-cancellativity, and cancellativity are preserved under the product and coproduct constructions (Propositions 4.8 and 5.5).","tokens_in":20192,"tokens_out":18600,"duration_ms":154685,"significance":"The paper offers a new and coherent categorical framework for atomic monoids, with explicit descriptions of all limits and colimits that are genuinely new. The length-set formulas, especially Theorem 5.7 expressing the system of length sets of a product as the intersection of the factor systems, are concrete and potentially applicable to realization problems in factorization theory. The proofs are largely detailed and self-contained, and the authors correctly emphasize that AtoMon is not a full subcategory of Mon. The reliance on [18, Lemma 2.2] for Dedekind-finiteness of atomic monoids is an established background result and does not create circularity. Overall, if the missing verification in Proposition 6.1 is supplied, the paper makes a solid contribution.","major_comments":[{"comment":"In the coequalizer proof, the assertion \"it is immediate to check that if x is a non-unit, non-atom of K, then [x] is not an atom in Q\" is load-bearing and is not proven. The claim is needed not only to show that Q is atomic but also to establish the universal property: the induced homomorphism ψ: Q → M must be atom-preserving, so every atom of Q must be the class of an atom of K. The missing argument is short: since K is atomic, write x = a_1 ... a_n with n ≥ 2 and a_i ∈ A(K); then [x] = [a_1] [a_2 ... a_n] in Q, and by the unit characterization already established, [a_1] and [a_2 ... a_n] are non-units of Q (a product of atoms of K is a non-unit, and units of Q are exactly the classes of units of K). Hence [x] is a product of two non-units and is not an atom. The authors should include this verification.","section":"Proposition 6.1"}],"minor_comments":[{"comment":"In the definition of U_P, the condition \"f(x) = g(x)\" should read \"f(x) = g(y)\".","section":"Proposition 6.5"},{"comment":"The statement ends with \"for every i ∈ I\", which is extraneous; the equality is between U_k(P_H) and the intersection over all i ∈ I of U_k(H_i).","section":"Theorem 5.8"},{"comment":"In the proof, the claim that the index word i = i_{j_1} * ... * i_{j_m} belongs to Γ_H is dismissed as \"trivial but tedious to check\"; since Γ_H is the key combinatorial object in the formula, a brief justification of the three cases would improve readability.","section":"Theorem 4.10"},{"comment":"In part (3), the reduction to the case where the last letter of a is a non-unit is only indicated by reference to a similar argument in part (1); the cancellation step deserves a few explicit words.","section":"Proposition 4.8"},{"comment":"The display \"B_1 * ... * B_n = B'_1 * ... * B'_ℓ\" should refer to the reduced forms of the two words, not the original words themselves.","section":"Lemma 4.3"},{"comment":"The manuscript contains numerous typographical/OCR artifacts (e.g., \"/integerdivide\", \"/d47\", and the split \"CA TEGORY\" in the abstract) that should be corrected in the final version.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically sound except for the omitted verification in Proposition 6.1, which is local and easily supplied. I view the required revision as minor in scope, but because the gap occurs in a load-bearing proof (cocompleteness), I recommend major_revision per the journal's guidelines. The authors' reliance on [18, Lemma 2.2] is legitimate and does not create circularity. I expect that after the missing argument is added, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper delivers what it promises. It defines AtoMon, proves it complete and cocomplete, and gives explicit product/coproduct constructions that differ from those in Mon. The length-set formulas (Theorems 4.10, 4.11, 5.7, 5.8) are new and appear correct. I checked the central proofs—product, coproduct, equalizer, pullback, coequalizer—and found no serious gap.\n\nThe genuinely new content is the category itself and the computation of limits/colimits. The product construction (generated by all-unit and all-atom tuples) is a real departure from the direct product, and the coproduct being the free product is clean. The arithmetic applications are a bonus; they give a way to build monoids with prescribed length-set systems, which is directly relevant to realization problems.\n\nThe one soft spot is Proposition 6.1, the coequalizer proof. The text says 'it is immediate to check' that a non-unit non-atom of K does not become an atom in the quotient Q. That claim is load-bearing: without it, the induced map from the coequalizer need not preserve atoms, so the universal property in AtoMon would fail. The missing argument is short—since K is atomic, write x = uv where u and v are non-units; the unit-characterization already established in the proposition shows [u] and [v] are non-units in Q, so [x] is a product of two non-units and hence not an atom. Supplying this line completes the proof. It is a minor gap in presentation, not a flaw in the result.\n\nThe reliance on [18, Lemma 2.2] (Fan–Tringali) for Dedekind-finiteness of atomic monoids is fine: that is an independent published result, not an input to the main conclusion. Citation pattern is otherwise standard.\n\nWho is this for? Factorization theorists who want categorical tools, and category theorists curious about non-full subcategories of algebraic categories. A serious referee should engage with it; the proofs are detailed enough that a referee can verify the constructions in an afternoon. Recommend accept after a minor revision asking the authors to fill the omitted verification in Proposition 6.1.","headline":"Clean, useful category-theoretic toolkit for atomic monoids; completeness/cocompleteness and length-set formulas hold up, with one small omitted verification in the coequalizer proof that is easy to fill.","tokens_in":20759,"tokens_out":2901,"would_cite":true,"duration_ms":77982,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18A05","18A30","20M10","20M13","13A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A category of atomic monoids has all limits and colimits, with explicit product and coproduct constructions.","keywords":["atomic monoids","atom-preserving homomorphisms","coproducts","products","limits and colimits","length sets","factorization theory","free products"],"falsifier":"Test the coequalizer claim by searching finite atomic monoids for a pair of atom-preserving homomorphisms whose coequalizer quotient in the category of all monoids is not atomic; the pair from the free monoid on one generator to the free monoid on two generators sending the generator to two different generators yields an atomic quotient, and any pair yielding a non-atomic quotient would refute Proposition 6.1 and Theorem 6.3.","tokens_in":19752,"feed_emoji":"⚛️","tokens_out":12749,"duration_ms":110676,"temperature":0.7,"pith_summary":"The paper defines AtoMon, the category whose objects are atomic monoids—monoids in which every non-unit is a product of atoms—and whose morphisms are monoid homomorphisms that send atoms to atoms. Its main theorem is that AtoMon is complete and cocomplete: every small diagram has a limit and a colimit. The paper computes these constructions explicitly: coproducts and coequalizers coincide with those in the larger category of all monoids, while products and equalizers are new submonoid constructions. It also proves formulas for factorization lengths: in a product, an element's length set is the intersection of its coordinate length sets, and in a coproduct, length sets decompose as sums over index blocks. These formulas make the universal constructions usable for building monoids with prescribed arithmetic behavior.","feed_headline":"Atomic monoids get all limits and colimits","feed_subtitle":"New product and coproduct formulas let builders combine atomic monoids and predict their factorization lengths.","key_machinery":"The machinery that carries the argument is a pair of explicit monoid constructions plus their unit and atom characterizations. For coproducts, the free product $C_H$ of the family, built from reduced words over the disjoint union of the factors, has units exactly the words whose letters are all units and atoms exactly the unit-conjugated single atoms of one factor (Lemmas 4.4 and 4.5). For products, the monoid $P_H$ generated inside the direct product by the all-units tuples $U_H$ and the all-atoms tuples $A_H$ has units exactly $U_H$ and atoms exactly $A_H$ (Lemma 5.3). These characterizations do the work: they show the constructed monoids are atomic, that the structural maps preserve atoms, and that factorizations in the constructions reduce to factorizations in the factors, which yields the length-set formulas.","core_discovery":"On the paper's own terms, the central discovery is that the natural categorical framework for atomicity is well behaved: AtoMon has all limits and all colimits. Coproducts are the usual free products (Theorem 4.6), and every colimit is computed as in the category of all monoids (Theorem 6.3). Products, however, are not direct products: the product of a family is the submonoid of the direct product generated by tuples that are all units or all atoms (Theorem 5.4), and equalizers are submonoids generated by the atoms and units on which two maps agree (Proposition 6.4). Along the way the paper characterizes the units and atoms of the free product and of this product submonoid, and derives arithmetic consequences: the length set of an element in a product is the intersection of the length sets of its components (Proposition 5.6), the system of length sets of a product is the componentwise intersection of the factors' systems (Theorem 5.7), and the non-zero length sets of a coproduct are exactly finite sums of factor length sets indexed by words with no redundant repetitions (Theorem 4.10).","pith_inferences":["These constructions could be used to attack realizability questions for systems of length sets, since the product and coproduct formulas turn the problem of engineering a monoid with a given system into a problem about intersections and finite sums of numerical sets.","The fact that products differ from direct products while coproducts agree with free products suggests a duality: colimits respect atomicity automatically, while limits require an explicit repair; this asymmetry may be worth testing on other categories defined by a closure property of the same kind.","One testable extension is to replace atoms by irreducibles in the sense of factorable monoids, as the paper itself proposes; the same limit and colimit questions would determine whether the categorical behavior found here is a special feature of atomicity or a general phenomenon.","The terminal object, a three-element monoid, is a stripped-down atom detector; using it as a target, any atomic monoid has a canonical morphism, so the paper's constructions could be used to classify atomic monoids by how their non-atom elements multiply."],"forward_implications":["Every diagram of atomic monoids has a limit and a colimit; colimits are the same as in the category of all monoids, so free products and quotients of atomic monoids by congruence pairs remain atomic.","The coproduct of any family of atomic monoids is the free product, so one can combine atomic monoids without changing the colimit behavior familiar from the larger category of monoids.","The product construction gives a way to form new atomic monoids whose length sets are intersections of given systems of length sets, making systems of length sets a resource that can be combined componentwise.","The union-of-length-sets invariant $U_k$ of a coproduct is computed from the factors' $U_k$ by summing over index words, which gives a recursive handle on unions of length sets.","The atom functor from AtoMon to sets is left adjoint to the free-monoid functor, yet it does not preserve coproducts, so it has no right adjoint."],"supporting_citations":[{"why":"Supplies the free product construction and monoid congruences used for the coproduct and coequalizer computations.","marker":"[26]"},{"why":"Supplies the theorems that products plus equalizers give completeness and coproducts plus coequalizers give cocompleteness, plus the pushout construction.","marker":"[5]"},{"why":"Provides the Dedekind-finiteness and unit/atom facts for atomic monoids that the unit and atom characterizations rely on.","marker":"[18]"},{"why":"Supplies the standard theory of non-unique factorization, length sets, and the terminology the paper extends.","marker":"[20]"},{"why":"Provides a realization result for systems of length sets that motivates the arithmetic formulas for products and coproducts.","marker":"[22]"}],"fun_headline_variants":["Atomic monoids: all limits and colimits exist","Atomic monoids are complete and cocomplete","Atomic monoids: universal constructions and length sets","Atomic monoids: products and coproducts solved","All limits and colimits for atomic monoids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of cocompleteness depends on the assertion, made without proof, that in the quotient used to build coequalizers a non-unit that was a product of several atoms cannot become a single atom; if that assertion is wrong, atomicity of the coequalizer fails and the cocompleteness theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Atomic monoids: all limits and colimits exist","Atomic monoids are complete and cocomplete","Atomic monoids: universal constructions and length sets","Atomic monoids: products and coproducts solved","All limits and colimits for atomic monoids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000421,"raw_usage":{"total_tokens":2120,"prompt_tokens":857,"completion_tokens":1263,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":1187}},"tokens_in":473,"tokens_out":1263,"duration_ms":28210,"temperature":1.0,"reasoning_tokens":1187,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T14:56:11.896943+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the coequalizer claim by searching finite atomic monoids for a pair of atom-preserving homomorphisms whose coequalizer quotient in the category of all monoids is not atomic; the pair from the free monoid on one generator to the free monoid on two generators sending the generator to two different generators yields an atomic quotient, and any pair yielding a non-atomic quotient would refute Proposition 6.1 and Theorem 6.3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the free product construction and monoid congruences used for the coproduct and coequalizer computations."},{"cited_title":"Borceux, Handbook of categorical algebra 1 , Cambridge University Press, 1994","cited_arxiv_id":null,"evidence_quote":"Supplies the theorems that products plus equalizers give completeness and coproducts plus coequalizers give cocompleteness, plus the pushout construction."},{"cited_title":"Fan and S","cited_arxiv_id":null,"evidence_quote":"Provides the Dedekind-finiteness and unit/atom facts for atomic monoids that the unit and atom characterizations rely on."},{"cited_title":"Geroldinger and F","cited_arxiv_id":null,"evidence_quote":"Supplies the standard theory of non-unique factorization, length sets, and the terminology the paper extends."},{"cited_title":"Geroldinger and Q","cited_arxiv_id":null,"evidence_quote":"Provides a realization result for systems of length sets that motivates the arithmetic formulas for products and coproducts."}],"review_version":1}