{"id":"45f6eb9d-1cab-438e-b006-b4754bfbab46","arxiv_id":"2506.15622","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Cyclic dendroidal sets and cyclic dendroidal spaces admit model structures for cyclic quasi-operads and complete Segal spaces, both Quillen equivalent to simplicial cyclic operads.","lead":"This paper constructs model structures for cyclic dendroidal sets and spaces, capturing cyclic infinity-operads. It proves both are Quillen equivalent to the established model for simplicial cyclic operads, resolving a 2021 conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted proof of Exercise 4.19 leaves the EZ/skeletal structure of Υ unverified; the boundary/skeleton arguments under Prop. 4.25 and Prop. 7.13 depend on it.","rationale":"The central claim, Theorem B, was traced through its main dependencies: the right-induced model structure of Section 6, the Beck–Chevalley induction of Proposition 7.13, and the Quillen equivalence lifting of Theorem 7.14. Within the main proof, I found the arguments for Proposition 6.4, Lemmas 7.10 and 7.12, and the application of Theorem A.2 internally coherent; the natural transformation β is indeed determined on representables because it is a natural transformation between colimit-preserving functors, and the induction using ∂ΥT is sound if the skeletal decomposition holds. The single unproved combinatorial input is Exercise 4.19, which asserts the hereditary EZ/skeletal properties for Υ. This is load-bearing because Proposition 4.25 and the identification of boundaries with skeleta are used in the induction, and the absolute pushout property is needed for the EZ formalism behind the Reedy identifications. However, the transfer from Ω to Υ is very plausible via the discrete fibration/opfibration properties, and the reader's CONDITIONAL verdict already accounts for this missing proof. I found no internal inconsistency or evidence that the claim is false; the concern is that a necessary supporting lemma is left as an exercise rather than being proved or referenced. Thus the verdict should remain conditional rather than being strengthened or overturned.","tokens_in":34665,"tokens_out":38950,"duration_ms":387415,"concrete_test":"Give a complete proof of Exercise 4.19 for Υ, or supply a citation for the hereditary EZ/skeletal structure. Concretely, for arbitrary negative maps φ:S→T and ψ:S→T′ in Υ, choose a root r∈bd(S), lift both maps along f to Ω using Proposition 4.16, form the absolute pushout in Ω, and verify that the resulting cospan in Υ is a pushout independent of the chosen root and is preserved by every functor. If the independence or absoluteness check fails for any pair of trees with at most four vertices, the boundary/skeleton arguments in Proposition 4.25 and Proposition 7.13 collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Exercise 4.19 states that four properties of Ω transfer to Υ: positive maps are monomorphisms, negative maps are split epimorphisms, negative maps are determined by their sections, and pairs of negative maps with common domain admit absolute pushouts. It concludes that Υ is a catégorie squelettique and a Berger–Moerdijk EZ category. No proof is supplied, and the transfer is not a formal consequence of the stated Propositions 4.12 and 4.16 alone: absolute pushouts are not preserved by arbitrary functors, so one must actually construct them in Υ or lift to Ω and verify independence of choices, not merely cite that f is a discrete fibration/opfibration. These properties are load-bearing. Proposition 4.25 uses Cisinski 8.1.35 to identify normal monomorphisms as the saturation of boundary inclusions, which requires the skeletal-category structure; [BM11, Cor. 6.10] is used in the induction in Proposition 7.13 to identify ∂ΥT with the (n−1)-skeleton; and the absolute pushout property underpins the EZ formalism used in the Reedy identifications of Section 8. If, for example, a pair of negative maps S→T and S→T′ in Υ had a pushout that was not absolute, then f would not preserve the Ω-pushout and the skeletal decomposition of ∂ΥT could fail, removing the generating set of cofibrations for the right-induced model structure. The gap is probably fillable, but it is asserted as an exercise rather than proved or referenced for Υ, leaving the central scaffolding with an unverified combinatorial input.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs model structures on cyclic dendroidal sets and cyclic dendroidal spaces: Theorem A gives a model structure on bΥ whose fibrant objects are the cyclic quasi-operads, Theorem C gives the cyclic dendroidal Rezk model structure on sbΥ, and Theorems B and D show that these are Quillen equivalent to the existing model structure on simplicial cyclic operads. Theorem E extends the statement to planar cyclic and planar non-cyclic operads, with the planar cyclic case intended to complete Walde's comparison with cyclic 2-Segal spaces. The main technical device is the root-elision functor f: Ω → Υ, used to right-induce model structures from the known dendroidal ones and to lift the Cisinski--Moerdijk Quillen equivalence via a Beck--Chevalley condition.","tokens_in":35026,"tokens_out":9206,"duration_ms":103768,"significance":"If the arguments are correct, the paper resolves the Drummond-Cole--Hackney conjecture and establishes model-independence across the cyclic quasi-operad, simplicial cyclic operad, and cyclic dendroidal Rezk settings. The paper contains substantial and mostly detailed proofs: Lemma 6.2 is a careful pushout computation, Proposition 7.13 is an induction over skeleta, and Theorem 8.2 is a nontrivial application of the localization calculus in Appendix A. The planar consequences are a genuine extra contribution. The main weakness is that a load-bearing combinatorial assertion, Exercise 4.19 on the skeletal/EZ structure of Υ, is left unproved; because Proposition 4.25, the boundary/skeleton identifications, and the induction in Proposition 7.13 all depend on it, the central scaffolding is not yet fully verified. The gap appears fillable, but it must be closed in the manuscript.","major_comments":[{"comment":"Exercise 4.19 asserts that the four properties of Ω transfer to Υ and that Υ is consequently a catégorie squelettique and a Berger--Moerdijk EZ category, but no proof is supplied. This is not a harmless exercise: Proposition 4.25 invokes Cisinski 8.1.35 to identify normal monomorphisms as the saturation of boundary inclusions, and Definition 4.22 uses [BM11, Corollary 6.10] for the equality ∂ΥT = sk_{n-1}ΥT, both of which require the skeletal/EZ structure. The transfer of property (4), absolute pushouts, is not a formal consequence of Propositions 4.12 and 4.16 alone: one must either construct the absolute pushout in Υ explicitly or prove that a pair of negative maps in Υ lifts to a common domain in Ω and that the image of the absolute Ω-pushout is a pushout in Υ. This argument, or a precise reference to it, must be included.","section":"Section 4, Exercise 4.19"},{"comment":"The characterization of cofibrations in the cyclic quasi-operad model structure as the normal monomorphisms depends on Proposition 4.25, which is stated without proof and whose cited source, Cisinski 8.1.35, applies only after Υ is known to be a catégorie squelettique. Since that skeletal structure is exactly what Exercise 4.19 is supposed to establish, Theorem A is currently contingent on an unverified combinatorial fact. The proof should either promote Exercise 4.19 to a theorem with a complete proof or replace Proposition 4.25 with a directly verified statement for Υ.","section":"Section 6, Theorem 6.5 and Proposition 4.25"},{"comment":"The induction proving the Beck--Chevalley isomorphism uses that ∂ΥT is the (n-1)-skeleton of ΥT for T of degree n, citing [BM11, Corollary 6.10]. That corollary applies in the EZ-categorical setting, and the EZ axioms are precisely the content of Exercise 4.19. Thus the proof of Proposition 7.13, and with it Theorem 7.14, is not fully justified until the missing skeletal/EZ argument is supplied. This is the same underlying gap as in the previous comments, but it is load-bearing at the exact point where the main Quillen equivalence is concluded.","section":"Section 7, Proposition 7.13"}],"minor_comments":[{"comment":"The sentence 'we next introduct an analogue' contains a typo; it should read 'we next introduce an analogue'.","section":"Section 7, before Definition 7.2"},{"comment":"The affiliation of the second author is given as 'University of Louisiana at Laf ayette'; this should be corrected to 'Lafayette'.","section":"Author affiliation"},{"comment":"The sentence 'by [BM11, Corollary 6.10] we have ∂ΥT = sk_{n-1}ΥT for T ∈ Υ of degree n' is stated as if the applicability of the cited corollary were already established; given the dependence on Exercise 4.19, this should be flagged explicitly at this point or moved after the skeletal structure is proved.","section":"Section 4, Definition 4.22"},{"comment":"In Corollary 7.8, the map γ∂ΩT,t is referenced without having been explicitly named beforehand; a short phrase identifying it as the component of γ at the boundary ∂ΩT,t would improve readability.","section":"Section 7, Lemma 7.7 and Corollary 7.8"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct and the omitted Exercise 4.19 is probably fillable by a short argument using absoluteness of the Ω-pushouts and the lifting properties of f. However, because the unproved exercise supports the cofibration description, the skeletal identifications, and the main Beck--Chevalley induction, I cannot recommend acceptance in the present form. I would be willing to look at a revised version in which Exercise 4.19 is replaced by a full proof or a complete reference. The reliance on theses for auxiliary statements (Ell23, Gag15) is acceptable but might be worth a sentence of guidance for readers."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read of Doherty–Hackney, arXiv:2506.15622. It's a strong paper that largely delivers what it claims: new model structures on cyclic dendroidal sets and cyclic dendroidal spaces, Quillen equivalent to the simplicial cyclic operad model, answering the Drummond-Cole–Hackney conjecture. The Beck–Chevalley step (Prop. 7.13) is genuinely substantial and proven in detail, and the pushout analyses in Lemmas 6.2 and 7.10 are careful. The planar consequences in Theorem E are a nice bonus, and the connection to Walde's 2-Segal program is made concrete.\n\nWhat is actually new: the model structures on bΥ and sbΥ, the rigidification equivalence with sCyc, and the planar omnibus theorem. The framework extends Cisinski–Moerdijk via known lifting theorems, but the cyclic combinatorics are new, and the target theorems do not appear in the earlier literature. The reliance on the authors' prior work is not circular; DCH21 and DCH19 are established independently.\n\nThe main soft spot is Exercise 4.19. It asserts, without proof, that four properties of Ω transfer to Υ, and concludes that Υ is a catégorie squelettique and a Berger–Moerdijk EZ category. This is load-bearing: Prop. 4.25 identifies normal monomorphisms as the saturation of boundary inclusions via Cisinski 8.1.35, which requires the skeletal-category structure; the induction in Prop. 7.13 uses ∂ΥT = sk_{n−1}ΥT via BM11 Cor. 6.10; and the Reedy identifications in Section 8 rely on the EZ formalism. The stress-test note is right that the transfer is not purely formal: absolute pushouts are not preserved by arbitrary functors, so one must actually construct them in Υ or verify independence of choices. This is probably fillable, but it is not a pedantic complaint. As written, the paper leaves a central combinatorial input as an exercise.\n\nMinor items: some auxiliary claims are delegated to a PhD thesis and a master's thesis (Ell23, Gag15). That is acceptable if those are accessible, but a referee should check that the cited results genuinely cover the cyclic case.\n\nBottom line: for anyone working in higher operads, dendroidal homotopy theory, or cyclic operads, this is a must-read. The main results are new, the proofs are mostly detailed, and the gap in Exercise 4.19 is real but likely repairable. It deserves serious peer review; the referee should ask for a proof or a precise reference for the transfer of the four properties to Υ, and for verification of the thesis citations.","headline":"Strong, new model-theoretic results for cyclic infinity-operads, with detailed main proofs; the unproved Exercise 4.19 leaves a load-bearing but likely repairable combinatorial gap.","tokens_in":35539,"tokens_out":2585,"would_cite":true,"duration_ms":26709,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18M85","18N40","55U35","18N70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The homotopy theory of cyclic infinity-operads has three equivalent presentations.","keywords":["cyclic operad","infinity-operad","dendroidal set","model category","Quillen equivalence","Segal space","cyclic dendroidal category","homotopy coherent nerve"],"falsifier":"Search the category of unrooted trees for an active tree map that does not admit a unique lift after a root is chosen, or an active map that factors through an outer coface; the paper's propositions assert neither happens, and the pushout decomposition in Lemma 6.2, which drives the rest of the argument, would fail if either did.","tokens_in":34480,"feed_emoji":"🔄","tokens_out":6946,"duration_ms":66097,"temperature":0.7,"pith_summary":"The paper seeks to establish that cyclic infinity-operads admit a well-behaved homotopy theory in three equivalent guises: cyclic dendroidal sets, cyclic dendroidal spaces, and simplicial cyclic operads. This matters because cyclic operads encode algebraic structures with no distinguished output, as in cyclic homology and topological field theory, and an infinity-categorical model lets homotopy-invariant versions of those structures be studied. The authors prove the required model structures exist and are Quillen equivalent, resolving a conjecture in the literature and completing the model-categorical foundation for connecting cyclic 2-Segal spaces to planar cyclic infinity-operads.","feed_headline":"Cyclic infinity-operad models are all Quillen equivalent","feed_subtitle":"Dendroidal sets, dendroidal spaces, and simplicial cyclic operads now share one homotopy theory.","key_machinery":"The load-bearing object is root-elision, the functor from rooted trees to unrooted trees that forgets the chosen root. The target category has as objects unrooted trees whose boundary is a set of arcs, and presheaves on it are cyclic dendroidal sets. The paper shows root-elision is a discrete fibration, meaning every tree map with a root in the codomain lifts uniquely; moreover, its restriction to active maps, those bijecting boundaries, is a discrete opfibration, and active maps are exactly those that do not factor through outer cofaces. These combinatorial facts produce the pushout decompositions of Lemma 6.2, which yield the right-induced model structure on cyclic dendroidal sets, the Quillen property of the associated adjoint string, and the Beck-Chevalley induction in Proposition 7.13 that upgrades the known rooted equivalences to the cyclic ones. The natural isomorphism between the cyclic and rooted rigidification functors provides the adjunction compatibility that this induction uses.","core_discovery":"The paper's central claim is that cyclic infinity-operads can be presented homotopically in three equivalent ways. It constructs a model structure on cyclic dendroidal sets whose fibrant objects are cyclic quasi-operads, meaning presheaves on the unrooted-tree category with fillers for all inner horns, and shows the homotopy coherent nerve and rigidification adjunction between this category and simplicial cyclic operads is a Quillen equivalence. It also constructs the cyclic dendroidal Rezk model structure on simplicial-valued presheaves whose fibrant objects are complete cyclic dendroidal Segal spaces, and proves the inclusion of ordinary cyclic dendroidal sets into these spaces is a left Quillen equivalence. The planar versions of all statements follow by slicing over the associative cyclic operad. Together these results affirmatively answer the question of whether a model structure for cyclic infinity-operads exists and is comparable to the known Dwyer-Kan model structure on simplicial cyclic operads.","pith_inferences":["A general template emerges: whenever a root-forgetting functor between tree categories is a discrete fibration and a discrete opfibration on active maps, the rooted model structures and equivalences should lift to the unrooted side; modular shape categories are a natural test case.","The planar results make it likely that cyclic 2-Segal spaces are equivalent, as infinity-categories, to invertible planar cyclic infinity-operads, since the restriction functor to Connes' cyclic category plus the new planar model structures supply the missing equivalence.","The Beck-Chevalley induction on skeleta suggests a reusable proof strategy for lifting Quillen equivalences along discrete fibrations, which could be tested on other families of generalized operads such as profinite or enriched variants.","If a cyclic Boardman-Vogt tensor product were constructed, one could check for a reverse Quillen equivalence from complete cyclic dendroidal spaces back to cyclic quasi-operads; the paper explicitly leaves this direction open."],"forward_implications":["The homotopy coherent nerve from simplicial cyclic operads to cyclic dendroidal sets becomes a right Quillen equivalence, so any infinity-categorical construction in one model transfers to the others.","Complete cyclic dendroidal Segal spaces give a second presentation of cyclic infinity-operads via Segal conditions plus Rezk completeness, and the inclusion of discrete objects is a Quillen equivalence.","Planar cyclic quasi-operads, planar simplicial cyclic operads, and planar cyclic dendroidal Rezk spaces are all Quillen equivalent, completing the bridge between cyclic 2-Segal spaces and invertible planar cyclic infinity-operads.","The model structure for cyclic quasi-operads is both right- and left-induced from the rooted dendroidal model structure, giving explicit generating cofibrations and a concrete description of cofibrant objects."],"supporting_citations":[{"why":"Introduces the cyclic dendroidal category and its Reedy structure, the shape category on which all the new presheaf categories are built.","marker":"[HRY20a]"},{"why":"Supplies the discrete fibration property of root-elision and the formula for the left Kan extension on objects, driving the pushout lemmas.","marker":"[Hac24a]"},{"why":"Establishes the Quillen equivalence between dendroidal sets and simplicial operads whose cyclic analogue is the paper's Theorem B.","marker":"[CM13b]"},{"why":"Defines the model structure for quasi-operads on dendroidal sets that is lifted to cyclic dendroidal sets.","marker":"[CM11]"},{"why":"Provides the dendroidal Rezk model structure and the discrete-to-space Quillen equivalence lifted in the paper's Theorem D.","marker":"[CM13a]"},{"why":"Gives criteria for right- and left-induced model structures and the theorem for lifting Quillen equivalences along adjoint strings.","marker":"[DCH19]"},{"why":"Constructs the Dwyer-Kan model structure on simplicial cyclic operads and poses the conjecture that the paper answers.","marker":"[DCH21]"},{"why":"Proves the natural isomorphism between cyclic and rooted rigidification functors, a key step in the Beck-Chevalley argument.","marker":"[Ell23]"},{"why":"Identifies cyclic 2-Segal spaces with invertible planar cyclic infinity-operads conditional on the planar model structures proven here.","marker":"[Wal21]"}],"fun_headline_variants":["All cyclic ∞-operad model structures are Quillen equivalent","Cyclic quasi-operads and Segal spaces: one homotopy theory","Quillen equivalences bridge cyclic dendroidal sets, spaces, and simplicial operads","Dendroidal, simplicial, and space models of cyclic ∞-operads are Quillen equivalent","Answering the cyclic ∞-operad model structure question with Quillen equivalences"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The chain of model structures and equivalences rests on root-elision: every map of unrooted trees with a chosen boundary element must lift uniquely to a map of rooted trees, and active maps, those that biject boundaries, must lift uniquely while never factoring through outer faces.","fun_headline_variants_meta":{"raw":{"variants":["All cyclic ∞-operad model structures are Quillen equivalent","Cyclic quasi-operads and Segal spaces: one homotopy theory","Quillen equivalences bridge cyclic dendroidal sets, spaces, and simplicial operads","Dendroidal, simplicial, and space models of cyclic ∞-operads are Quillen equivalent","Answering the cyclic ∞-operad model structure question with Quillen equivalences"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001166,"raw_usage":{"total_tokens":4771,"prompt_tokens":837,"completion_tokens":3934,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":453,"completion_tokens_details":{"reasoning_tokens":3820}},"tokens_in":453,"tokens_out":3934,"duration_ms":26553,"temperature":1.0,"reasoning_tokens":3820,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:32:08.193041+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search the category of unrooted trees for an active tree map that does not admit a unique lift after a root is chosen, or an active map that factors through an outer coface; the paper's propositions assert neither happens, and the pushout decomposition in Lemma 6.2, which drives the rest of the argument, would fail if either did.","supporting_citations":[],"review_version":2}