{"id":"943bd922-3cc7-41a0-9ac9-fa36d67b01b7","arxiv_id":"2606.30254","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends computads with invertible generators, constructs a coreflection to ordinary computads that preserves generated ω-categories, proves those ω-categories are cofibrant, and shows the subcategory of generalised computads with generator-preserving morphisms is a presheaf topos.","lead":"The paper extends computads for weak ω-categories to mark some generators as invertible and gives a finite description of the walking equivalences that classify invertible cells. Researchers working on higher categories or homotopy theory may use this to handle equivalences and cofibrant objects more directly.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption directly identifies the inductive well-definedness plus coreflection preservation as the critical step; the full-text description does not introduce any additional gap that would alter the UNVERDICTED status or raise a new load-bearing risk beyond what the reader already flagged.","tokens_in":1609,"tokens_out":257,"duration_ms":24523,"concrete_test":"Explicitly compute the coreflection on the walking equivalence (the 2-dimensional case with a single invertible 1-cell) and verify that the image under the coreflection functor yields an isomorphic ω-category to the original generalised-computad presentation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract and described construction outline an inductive free-generation process for generalised computads (with invertible generators) together with an explicit coreflection into ordinary computads that is claimed to preserve the generated ω-category. No internal inconsistency, circularity in the induction, or hidden assumption about dimension-wise well-definedness is detectable from the stated claims; the cofibrancy conclusion follows directly once the coreflection is established, and the presheaf-topos claim is a standard consequence of the generator-preserving morphisms.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper extends the notion of computads for weak ω-categories to generalised computads that allow certain generators to be marked as invertible. It gives an inductive description of the free ω-categories generated by these objects (including a finite presentation of the walking equivalences), constructs a coreflection from generalised computads to ordinary computads that preserves the generated ω-category, and concludes that the resulting ω-categories are cofibrant. It further shows that the subcategory of generalised computads equipped with generator-preserving morphisms forms a presheaf topos.","tokens_in":1691,"tokens_out":367,"duration_ms":28136,"significance":"If the inductive constructions and coreflection are correct, the work supplies a concrete, finite description of walking equivalences and establishes cofibrancy results that are useful for model-categorical approaches to weak ω-categories. The demonstration that the relevant subcategory is a presheaf topos is a clean categorical property that parallels known results for ordinary computads and may facilitate further structural investigations.","major_comments":[],"minor_comments":[{"comment":"The introduction refers to the 'corresponding result for ordinary computads' when stating the presheaf-topos property but does not cite the specific reference; adding this citation would improve traceability.","section":null},{"comment":"Notation distinguishing marked invertible generators from ordinary ones (introduced early in the definitions) would benefit from a short illustrative example immediately after the definition to clarify the marking convention before the inductive clauses begin.","section":null},{"comment":"The statement that the coreflection preserves the generated ω-category is central; a brief remark on how the preservation interacts with the dimension-wise inductive steps would aid readability even if the full verification appears later.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the paper, the significance statement, and the recommendation of minor revision. No major comments were provided in the report.","responses":[],"tokens_in":1147,"tokens_out":51,"duration_ms":16206,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core advance is marking some generators invertible inside the computad formalism for weak ω-categories. This produces an inductive description of the free ω-categories they generate and, crucially, a finite presentation of the walking equivalences. They then build a coreflection from these generalised computads back to ordinary ones that preserves the generated ω-category, which immediately yields the cofibrancy statement. The subcategory of generalised computads with generator-preserving maps is shown to be a presheaf topos, mirroring the ordinary case.\n\nThe constructions look like a direct, workable extension rather than a radical departure. The coreflection is the part that does real work: once it is in place, cofibrancy follows without extra machinery. The topos result is a standard consequence once the morphisms are fixed.\n\nThe inductive clauses for free generation with marked invertibles need to be checked dimension by dimension for well-definedness, but nothing in the stated claims suggests a circularity or hidden assumption that would break the induction. The abstract is terse on proof details, so a referee would want to see the explicit inductive clauses and the verification that the coreflection is indeed a right adjoint.\n\nThis is aimed at people already using computads to present weak higher categories or working on cofibrant replacements in homotopy theory. It is narrow but technically sharp, and the claims are specific enough to be worth referee time even if they require some rewriting for clarity.","headline":"The paper adds invertible generators to computads, gives a finite handle on walking equivalences, and uses a coreflection to get cofibrancy plus a topos structure.","tokens_in":2205,"tokens_out":369,"would_cite":false,"duration_ms":18239,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Marking certain generators as invertible in computads for weak ω-categories yields an inductive free construction, a coreflection to ordinary computads, and cofibrant generated categories.","keywords":["computads","weak ω-categories","invertible generators","walking equivalences","cofibrant objects","presheaf topos","coreflection"],"falsifier":"An explicit dimension at which the inductive construction of the free ω-category on a generalised computad fails to be well-defined, or a generalised computad whose coreflection image does not generate an isomorphic ω-category, would falsify the claims.","tokens_in":2506,"feed_emoji":"","tokens_out":680,"duration_ms":28352,"temperature":0.7,"pith_summary":"The paper extends computads for weak ω-categories so that some generators can be marked invertible. It gives an inductive description of the free ω-categories these generalised computads generate, including a finite presentation of the walking equivalences that classify invertible cells. A coreflection is built from the generalised computads to ordinary ones that preserves the generated ω-categories. This coreflection is used to conclude that ω-categories generated by generalised computads are cofibrant. The subcategory of generalised computads equipped with generator-preserving morphisms is shown to be a presheaf topos.","feed_headline":"Marking generators invertible yields cofibrant ω-categories","feed_subtitle":"Generalised computads admit a coreflection to ordinary ones that preserves generated categories and form a presheaf topos.","key_machinery":"Generalised computads with marked invertible generators, together with their inductive free ω-category construction and the coreflection to ordinary computads.","core_discovery":"Generalised computads mark selected generators as invertible and generate free weak ω-categories through an inductive construction that remains finite at each dimension. The construction admits a coreflection into ordinary computads that preserves the generated ω-category, from which it follows that the generated ω-categories are cofibrant. The category whose objects are generalised computads and whose morphisms preserve the generators is a presheaf topos.","pith_inferences":["The finite description of walking equivalences may simplify explicit calculations of invertible cells in higher-dimensional pasting diagrams.","If the same marking technique extends to other presentations of higher categories, cofibrancy results could transfer across different model structures.","The presheaf topos structure suggests that generalised computads support a well-behaved internal logic for reasoning about invertible cells."],"forward_implications":["Walking equivalences receive a simple finite description as free ω-categories on generalised computads.","ω-categories generated by generalised computads are cofibrant in the model structure under consideration.","The category of generalised computads with generator-preserving morphisms is a presheaf topos.","Properties of ordinary computads that are preserved by the coreflection transfer directly to the generalised setting."],"fun_headline_variants":["Generalised computads with invertible generators yield cofibrant ω-categories","Coreflection preserves ω-categories generated by generalised computads","Category of generalised computads is presheaf topos","Finite inductive free ω-categories from invertible computad generators"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The inductive description of the free ω-categories generated by the generalised computads is well-defined at every dimension and admits a coreflection to ordinary computads that preserves the generated ω-category.","fun_headline_variants_meta":{"raw":{"variants":["Generalised computads with invertible generators yield cofibrant ω-categories","Coreflection preserves ω-categories generated by generalised computads","Category of generalised computads is presheaf topos","Finite inductive free ω-categories from invertible computad generators"]},"model":"grok-4.3","cost_usd":0.007707,"raw_usage":{"total_tokens":3472,"prompt_tokens":563,"num_sources_used":0,"completion_tokens":66,"cost_in_usd_ticks":77074500,"prompt_tokens_details":{"text_tokens":563,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2843,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":563,"tokens_out":66,"duration_ms":47986,"temperature":1.0,"reasoning_tokens":2843,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T03:13:19.295991+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit dimension at which the inductive construction of the free ω-category on a generalised computad fails to be well-defined, or a generalised computad whose coreflection image does not generate an isomorphic ω-category, would falsify the claims.","supporting_citations":[],"review_version":1}