{"id":"496fdb84-b54d-4f19-9e3d-647dd0f2d70e","arxiv_id":"2607.27846","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Gabi-monads on skew-closed categories are exactly the data needed to lift internal homs to Eilenberg-Moore categories; a gabi-monad is Hopf iff its parametric mates are invertible.","lead":"This paper defines gabi-monads — monads carrying enough extra structure to lift 'internal hom' objects to their categories of algebras — and proves that they correspond exactly to skew-closed structures on those categories for which the forgetful functor is strict closed. It also gives a clean criterion for when a gabi-monad is a Hopf monad, extending earlier gabi-algebra theory beyond the linear case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central bijection (Thm 3.6) inherits its key lift-classification from [BSV24, Lemma 3.1] without independent proof; no error found, but this is the load-bearing verification gap.","rationale":"I read the paper in good faith and found no internal inconsistency, data manipulation, or post hoc selection. The new material — normal gabi-monads, parametric-mate characterization, dyadic gabi-monads, and the examples — appears to follow coherently from the stated machinery. The most load-bearing point is indeed the same one the Reader identified: the reconstruction theorem is inherited from [BSV24] rather than reproved, and the local proof of Theorem 3.6 relies on unexpanded diagram chases and the phrase 'easily seen to be inverse'. This is a verification gap, not a discovered falsehood. Because the imported lemmas are plausible, standard in flavor, and the surrounding framework is consistent, I would not change the ACCEPT verdict; I would, however, encourage a future version to include a fully expanded proof or a machine-checked formalization of Lemma 3.1 and Theorem 3.6. The concrete formalization check proposed above would settle whether the concern actually lands.","tokens_in":50252,"tokens_out":18017,"duration_ms":183994,"concrete_test":"Formalize Lemma 3.1 and Theorem 3.6 in a proof assistant such as Lean/mathlib or UniMath: define C, T, C^T, the internal-hom lifting, and prove the bijection between lifts of [-,-] and natural transformations s satisfying (9), including both inverse constructions and diagram (9). Then formalize the forward direction of Theorem 3.6: from a gabi-monad, construct the skew-closed structure on C^T with U^T strict closed, and conversely. If the formalization succeeds, the concern is settled; if it fails, the failing diagram pinpoints the gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's backbone is Theorem 3.6: gabi-monad structures on T correspond bijectively to skew-closed structures on C^T for which U^T is strict closed. The proof is not self-contained: Lemma 3.1 — imported from [BSV24, Lemma 3.1] — asserts that every lift of the internal-hom functor [-,-] to C^T is encoded by a natural transformation s : T[T-,-] -> [-,T-] satisfying diagrams (9), and Theorem 3.6 additionally leans on [BSV24, Theorem 3.4]. If Lemma 3.1 missed a class of lifts, or if the diagrams (10)-(11) in Definition 3.3 do not fully force the coherence maps i,j,Γ to lift, then the reconstruction bijection, Proposition 3.10, and all Section 6 examples lose their foundation. The paper compresses the inverse verification as 'easily seen to be inverse', and no machine-checked or fully expanded proof is supplied. This is a real verification risk rather than a detected internal contradiction; the rest of the paper appears internally consistent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces gabi-monads on skew-closed categories, generalizing the gabi-algebras of Berger–Saracco–Vercruysse to a monadic setting. A gabi-monad is a monad T on a skew-closed category C equipped with a T-algebra structure on the unit object and a natural transformation s:T[T-,-] -> [-,T-] satisfying lifting axioms. The central result, Theorem 3.6, states that gabi-monad structures on T are in bijective correspondence with skew-closed structures on the Eilenberg–Moore category C^T for which the forgetful functor U^T is strict closed. The paper then gives a Hopf criterion in terms of invertible parametric mates (Theorem 4.18), a dyadic/star-autonomous version (Theorem 5.8), and a range of examples and non-examples, including torsion-free modules over an integral domain, preorders inside reflexive digraphs, simplicial complexes, and pointed sets as a near-example.","tokens_in":50476,"tokens_out":15885,"duration_ms":154765,"significance":"If correct, the paper provides a monadic Tannaka–Krein reconstruction theorem for skew-closed categories, unifying and extending prior work on Hopf monads and gabi-algebras. The Hopf characterization in Theorem 4.18 is explicit and checkable, and the examples in Section 6 are genuinely informative, showing that normal gabi-monads need not be Hopf and giving a clean Prüfer-domain criterion for the torsion-free example. The paper is careful with coherence conditions and contains substantial diagrammatic work. Its main caveat is that the central bijection is inherited from [BSV24], so the reader must trust that published result or have it reproduced in the monadic skew-closed setting.","major_comments":[{"comment":"The paper's main reconstruction theorem is not proven in full. Lemma 3.1, which classifies all lifts of [-,-] to C^T by natural transformations s satisfying (9), is quoted from [BSV24, Lemma 3.1] without proof. Lemma 3.5 is justified only by 'a direct computation', and in Theorem 3.6 the inverse verification is summarized as 'easily seen to be inverse'. Since Proposition 3.10, Theorem 4.18, and all Section 6 examples depend on this bijection, this is a load-bearing verification gap. I do not claim an error; [BSV24] is published. However, the manuscript should either reproduce the classification and the inverse check in the monadic skew-closed setting, or state the precise theorem from [BSV24] that covers this generality and explain the adaptation. As written, the proof is not self-contained at the central point.","section":"Theorem 3.6 / Lemma 3.1"},{"comment":"The reduction of the Hopf criterion to the single plus-minus map beta is stated in one sentence: 'by taking V=k and M=(A,m_A), we conclude'. This uses the fact that the relevant natural transformations are determined by their component at a projective generator in each variable, because the functors involved are additive and preserve coproducts. This is plausible and likely true, but it is not immediate and is load-bearing for recovering the ring-theoretic equivalence between normal gabi-algebras and Hopf algebras. Please expand this step.","section":"Corollary 4.19"}],"minor_comments":[{"comment":"Typo: 'on may choose' should be 'one may choose'.","section":"Section 1"},{"comment":"The diagrams (1a)-(1e) are not labeled in the text, and some of the arrows in these diagrams are ambiguous. Consider adding explicit labels or a paragraph explaining the notational conventions.","section":"Definition 2.1"},{"comment":"The claim that the Eilenberg–Moore category for E={0,1} is the category of rectangular bands is stated without detailed proof. The reference to [Kim58] is helpful, but a short verification of the equivalence would improve readability.","section":"Example 3.16"},{"comment":"The notation pΣ_{xV,Wy} is introduced abruptly. Since this example is one of the advertised non-Hopf gabi-monads, a few more words about the simplicial-complex closed structure and why the monad is gabi would be useful.","section":"Remark 6.36"},{"comment":"In the pointed-sets example, it is worth saying explicitly that the internal-hom lift is functorial but φ0 is not an isomorphism, which is precisely why the maybe monad is not a gabi-monad.","section":"Section 6.5"}],"recommendation":"major_revision","confidential_remarks":"The dependence on [BSV24] is substantial and two of the three authors of this manuscript are also authors of [BSV24]. The editor may wish to confirm that [BSV24, Theorem 3.4] indeed proves the monadic skew-closed statement used here, since the manuscript's main theorem is essentially a rephrasing of that result. The claimed new contributions—Theorem 4.18, Theorem 5.8, and the examples—are the strongest independent parts of the paper; the proof of Theorem 3.6 should be made self-contained or its import clearly delimited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper introduces gabi-monads as the monadic analogue of gabi-algebras on skew-closed categories. The genuinely new material is in Section 4 and Section 6: the characterization of Hopf monads via invertibility of parametric mates (Theorem 4.18), the dyadic *-autonomous version (Section 5), and a supply of normal-but-not-Hopf examples, especially torsion-free modules over integral domains and preorders as reflective subcategories. These are well chosen and internally consistent. The examples do real work: the torsion-free case showing that normality does not force Hopf, with a precise Prüfer-domain criterion, is the kind of concrete payoff that makes the framework worth having.\n\nThe soft spot is exactly what the stress-test note flags. Theorem 3.6, the backbone of the paper, is explicitly a rephrasing of [BSV24, Theorem 3.4], and Lemma 3.1 — which classifies lifts of the internal hom — is imported from the same source. That means the main bijection rests on a published result from the same research line, not on a proof supplied here. I see no sign of circularity or error: the cited theorem is parameter-free and its assumptions do not include the paper's conclusions. Still, the \"easily seen to be inverse\" step in Theorem 3.6 is exactly where a subtle failure of bijectivity would hide, and a self-contained proof or a formalized diagram chase would materially reduce the residual risk.\n\nWhat the paper does well beyond the new examples is the framing. The parametric mate criterion is a clean way to separate Hopf from merely gabi, and the paper is honest about which parts are imported. The prose is careful, and the category theory is standard for the area.\n\nWho is this for? People working on monads, Hopf monads, and reconstruction of closed categories. It deserves a serious referee, not a desk reject. If I were the editor, I would send it out with a request to either expand the proof of Theorem 3.6 or state explicitly which parts are cited and why the cited result applies unchanged. The referee should focus on the imported lemma and the inverse verification.","headline":"A competent extension of gabi-algebra theory to arbitrary skew-closed categories, but the central reconstruction theorem is cited from [BSV24] rather than proved here, so the genuinely new material is the Hopf criterion and the examples.","tokens_in":50997,"tokens_out":2290,"would_cite":false,"duration_ms":25833,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18C15","18D15","18C20","18M50","16T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that gabi-monad structures on a monad correspond bijectively to skew-closed structures on its Eilenberg–Moore category for which the forgetful functor is strict closed.","keywords":["gabi-monads","skew-closed categories","Eilenberg-Moore category","Hopf monads","reconstruction","internal hom","gabi-algebras","Tannaka-Krein"],"falsifier":"Exhibit a monad T on a skew-closed category C and a skew-closed structure on C^T for which the forgetful functor is strict closed but the natural transformation s defined by equation (14) fails one of the diagrams (9). Such a counterexample would directly contradict the bijection in Theorem 3.6.","tokens_in":50092,"feed_emoji":"🧮","tokens_out":4258,"duration_ms":42617,"temperature":0.7,"pith_summary":"The paper develops a Tannaka–Krein style reconstruction for closed categories, not just monoidal ones. It introduces gabi-monads—monads equipped with a compatibility map s and a unit algebra structure—and proves that these are exactly the structures needed to lift the internal hom functor to the category of algebras. The central bijection says that a gabi-monad structure is the same datum as a skew-closed structure on the Eilenberg–Moore category making the forgetful functor strict closed. From this, the paper characterizes when a gabi-monad is Hopf via invertibility of certain parametric mates, recovering and sharpening known facts about gabi-algebras. A rich family of examples shows that normal gabi-monads need not be Hopf.","feed_headline":"Gabi-monads exactly match lifts of internal homs","feed_subtitle":"A monad is gabi exactly when the internal hom lifts to its algebra category; Hopf monads are the invertible-mate case.","key_machinery":"The central object is a gabi-monad: a monad T on a skew-closed category C equipped with a T-algebra structure a_1:T1→1 on the closed unit and a natural transformation s:T[T-,-] → [-,T-] that satisfies the lifting compatibility diagrams (9). This data is precisely what is needed to make the internal hom functor lift to the Eilenberg–Moore category, with the action on hom-objects given by a_M * a_N := [M,a_N] ∘ s_{M,N} ∘ T[a_M,N]. The parametric mates of the associated maps γ^M_X then control whether the lifted closed structure is part of a tensor–hom adjunction, i.e., whether the gabi-monad is actually a left Hopf monad.","core_discovery":"The authors establish that for a monad T on a skew-closed category C, giving T a gabi-monad structure (a T-algebra structure on the unit plus a natural transformation s:T[T-,-] → [-,T-] satisfying compatibility diagrams) is bijectively equivalent to giving the Eilenberg–Moore category C^T a skew-closed structure such that the canonical forgetful functor U^T:C^T→C is strict closed. They then show that T is a left Hopf monad exactly when, for every algebra M, the mate of the map γ^M_X := s_{M,X} ∘ T[a_M, X]: T[M,X] → [M,TX] is invertible; this proves that normal gabi-algebras over a commutative ring are Hopf algebras. They also formulate a dyadic version for *-autonomous categories (where Hopf","pith_inferences":["The bijection suggests that 'closed' structure can be reconstructed before monoidal structure; one could test whether analogous lift criteria hold for other 2-categorical structures, e.g., lax monoidal functors between skew-monoidal categories when the unit is not lifted.","The pointed-set quasi-example shows that the unit-algebra condition is essential; a variant that drops this condition might recover many more lifts, a direction the authors flag as future work.","The invertibility criterion for Hopfness could serve as a concrete computational test in algebraic categories: one checks whether explicitly given s maps yield invertible mates, rather than searching for antipodes directly."],"forward_implications":["Every left Hopf monad on a closed monoidal category is a normal gabi-monad, so the notion genuinely extends Hopf monad theory.","The converse fails: torsion-free modules over a non-Prüfer integral domain and the transitive-closure reflection on reflexive digraphs give normal gabi-monads that are not Hopf.","In *-autonomous categories, a monad admits a Hopf structure if and only if it admits a normal dyadic gabi-monad structure, so the two notions coincide in that setting.","The ring-theoretic result that every normal gabi-algebra over a commutative ring is a Hopf algebra follows from the monadic characterization with a short, conceptual proof.","The reconstruction theorem gives a new route to detecting when a closed structure on a category of algebras comes from a monad on the base category."],"fun_headline_variants":["Gabi-monads = closed structures on EM category","Monad is gabi iff internal hom lifts","Hopf = gabi with invertible mates","Gabi-monads: Hopf algebras meet reflexive digraphs","Closed structure on algebras iff gabi-monad"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole reconstruction rests on the lemma (inherited from an earlier paper) that every lift of the internal-hom functor to the Eilenberg–Moore category is encoded by a natural transformation s of the specified form; if some lift could exist outside this form, the claimed bijection would not hold.","fun_headline_variants_meta":{"raw":{"variants":["Gabi-monads = closed structures on EM category","Monad is gabi iff internal hom lifts","Hopf = gabi with invertible mates","Gabi-monads: Hopf algebras meet reflexive digraphs","Closed structure on algebras iff gabi-monad"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1167,"prompt_tokens":766,"completion_tokens":401,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":325}},"tokens_in":510,"tokens_out":401,"duration_ms":4115,"temperature":1.0,"reasoning_tokens":325,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T00:20:08.427996+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a monad T on a skew-closed category C and a skew-closed structure on C^T for which the forgetful functor is strict closed but the natural transformation s defined by equation (14) fails one of the diagrams (9). Such a counterexample would directly contradict the bijection in Theorem 3.6.","supporting_citations":[],"review_version":1}