{"id":"4db37542-db9e-4eba-b034-db1af2a11907","arxiv_id":"2606.10052","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The span functor from double ∞-categories to ∞-categories admits a right adjoint given by squares, yielding new proofs of equivalences among the Q-, S-, cobordism, and squares models of algebraic K-theory.","lead":"The paper shows that the span construction on double infinity-categories has a right adjoint given by the category of squares. This adjunction supplies new proofs that several standard models of algebraic K-theory are equivalent.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flagged the functoriality-plus-adjoint-existence step as the key assumption. Because the supplied abstract contains no further technical detail that would allow detection of a flaw in that step, and because the instruction permits an honest non-finding, the verdict remains UNVERDICTED with no adjustment warranted.","tokens_in":1604,"tokens_out":258,"duration_ms":14449,"concrete_test":"Extract the precise statement of the adjunction (likely Theorem 1 or the main result in §3) and verify that the unit and counit satisfy the triangle identities after applying the constructions to the terminal double ∞-category; if both identities hold up to equivalence, the existence of the right adjoint is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states that the span construction is realized as a functor double-∞-Cat → ∞-Cat admitting a right adjoint given by the squares construction, and that this yields new proofs of known equivalences among K-theory models. No internal inconsistency, missing coherence datum, or circularity is detectable from the given statement; the claim is of a form that is standard to establish via universal properties in the ∞-categorical literature.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript establishes a universal property for the span ∞-category by realizing the span construction as a functor from double ∞-categories to ∞-categories that admits a right adjoint, with the right adjoint given by the double ∞-categories of squares. This adjunction is then applied to derive new proofs of the known equivalences among the Q-construction, S-construction, cobordism model, and squares construction of algebraic K-theory.","tokens_in":1689,"tokens_out":304,"duration_ms":15778,"significance":"If the central adjunction holds, the result supplies a clean universal-property description of functors out of the span ∞-category and furnishes alternative, non-circular derivations of the equivalences between several standard models of algebraic K-theory. Such an adjunction is a natural and potentially reusable tool in the ∞-categorical literature on K-theory.","major_comments":[],"minor_comments":[{"comment":"The abstract states the main theorem but does not indicate where in the text the functoriality of the span construction (double ∞-Cat → ∞-Cat) is verified or where the unit and counit of the adjunction are constructed; adding explicit section references would improve readability.","section":null},{"comment":"Notation for double ∞-categories and the squares construction should be introduced with a short preliminary subsection, as readers may encounter varying conventions in the ∞-categorical K-theory literature.","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 manuscript and for recommending minor revision. No major comments appear in the report, so we have no specific points requiring rebuttal or revision at this stage.","responses":[],"tokens_in":1082,"tokens_out":50,"duration_ms":6594,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that they treat the span construction as a functor from double ∞-categories to ∞-categories and exhibit the squares construction as its right adjoint. This adjunction then supplies fresh proofs that the Q-construction, S-construction, cobordism model, and squares model all give equivalent algebraic K-theory spaces.\n\nThe adjunction itself looks like the genuinely new piece. The equivalences were already in the literature, but the uniform argument via a single universal property is not. If the construction goes through cleanly, it could shorten some existing comparisons and make extensions to other models easier.\n\nThe approach is direct: define the functor, produce the adjoint, then derive the equivalences from the adjunction rather than the other way around. That avoids the circularity worry. The abstract is high-level, so the real test is whether the ∞-categorical coherence data for the functor and the adjunction are handled without gaps; nothing in the statement flags an obvious problem, and this style of argument is standard in the area.\n\nThe paper is for people already working with double ∞-categories and the standard models of algebraic K-theory. A reader who knows the prior equivalences will see the value in the new route; someone outside that circle will find it technical and narrow. The central claim is concrete and formally grounded enough to merit referee time, even if the proofs need polishing.\n\nI would send it to peer review.","headline":"The paper builds a span-squares adjunction on double ∞-categories and uses it for new proofs of known K-theory model equivalences.","tokens_in":2137,"tokens_out":361,"would_cite":false,"duration_ms":12983,"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":"The span construction, as a functor from double ∞-categories to ∞-categories, admits a right adjoint given by squares.","keywords":["span construction","double ∞-categories","algebraic K-theory","adjunction","squares construction","Q-construction","S-construction","cobordism model"],"falsifier":"An explicit functor out of the span ∞-category that cannot be obtained from any double ∞-category of squares would show the right adjoint does not exist.","tokens_in":2506,"feed_emoji":"","tokens_out":636,"duration_ms":14900,"temperature":0.7,"pith_summary":"The paper shows that the span construction can be viewed as a functor from double ∞-categories to ∞-categories and that this functor has a right adjoint defined by double ∞-categories of squares. This adjunction gives a universal property for the span ∞-category by describing functors out of it in terms of square data. As a direct consequence the adjunction supplies new proofs that the Q-construction, S-construction, cobordism model, and squares construction all yield equivalent versions of algebraic K-theory. A reader would care because the single adjunction replaces separate verifications of equivalence between each pair of models.","feed_headline":"Span functor from double ∞-categories admits squares right adjoint","feed_subtitle":"The adjunction supplies new proofs that Q-, S-, cobordism and squares constructions give equivalent algebraic K-theory.","key_machinery":"The span-squares adjunction, relating the span functor on double ∞-categories to the right adjoint given by double ∞-categories of squares.","core_discovery":"We view the span construction as a functor from double ∞-categories to ∞-categories and show that this functor admits a right adjoint defined by the double ∞-categories of squares. The resulting adjunction yields a universal property of the span ∞-category that describes its functors. Using the adjunction we obtain new proofs of the equivalences between the Q-, the S-, the cobordism model, and the squares construction of algebraic K-theory.","pith_inferences":["The same adjunction pattern may apply to other universal constructions that arise from double categories.","One could test whether analogous right adjoints exist when the base is replaced by other variants of ∞-categories."],"forward_implications":["Functors defined on the span ∞-category correspond to maps into double ∞-categories of squares.","The Q-construction of algebraic K-theory is equivalent to the S-construction via the adjunction.","The cobordism model of algebraic K-theory is equivalent to the squares construction via the adjunction.","Equivalences among all four listed models of algebraic K-theory follow from a single adjunction rather than pairwise comparisons."],"fun_headline_variants":["Span functor admits squares right adjoint","Squares define right adjoint to span functor","Span-squares adjunction proves K-theory equivalences","Universal property of spans via squares adjunction"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The span construction can be realized as a functor from double ∞-categories to ∞-categories that admits a right adjoint.","fun_headline_variants_meta":{"raw":{"variants":["Span functor admits squares right adjoint","Squares define right adjoint to span functor","Span-squares adjunction proves K-theory equivalences","Universal property of spans via squares adjunction"]},"model":"grok-4.3","cost_usd":0.00387,"raw_usage":{"total_tokens":1847,"prompt_tokens":544,"num_sources_used":0,"completion_tokens":51,"cost_in_usd_ticks":38703000,"prompt_tokens_details":{"text_tokens":544,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1252,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":544,"tokens_out":51,"duration_ms":6879,"temperature":1.0,"reasoning_tokens":1252,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T13:49:49.112302+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit functor out of the span ∞-category that cannot be obtained from any double ∞-category of squares would show the right adjoint does not exist.","supporting_citations":[],"review_version":1}