{"id":"3e5629b3-cd12-46a4-8888-ab701a9d6379","arxiv_id":"2409.19061","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":2.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Unifies decomposition spaces and 2-Segal spaces via active-inert factorization, path space criterion, and edgewise subdivision, with examples from outer face complexes.","lead":"The paper defines decomposition spaces using the active-inert factorization system on the simplicial category and shows their equivalence to 2-Segal spaces. A generalist might read it to see how two viewpoints on simplicial structures in category theory are connected.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flags the factorization-to-2-Segal equivalence as the load-bearing point, but that equivalence is established in the existing literature and matches the criteria listed in the abstract. No internal inconsistency or missing justification is present.","tokens_in":1566,"tokens_out":274,"duration_ms":10517,"concrete_test":"Take the standard 2-Segal pullback square for the map X_2 → X_1 × X_1 and verify it coincides with the active-inert pullback square induced by the factorization of [2] → [1] in Δ; check agreement on the three face maps.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the equivalence between decomposition spaces (via the active-inert factorization system on Δ) and 2-Segal spaces. This equivalence is a standard result in the literature (Gálvez-Carrillo–Kock–Tonks and Dyckerhoff–Kapranov), with the factorization system well-defined (active maps preserve endpoints, inert maps are order-preserving inclusions) and the pullback conditions matching the 2-Segal maps exactly. The path-space and edgewise-subdivision criteria are known to be equivalent characterizations; the paper is expository and introduces no new unsupported steps.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper provides an introduction to decomposition spaces and 2-Segal spaces, defining decomposition spaces via the active-inert factorization system on the simplicial category Δ and establishing their equivalence to 2-Segal spaces. It presents the path space criterion (characterizing decomposition spaces via upper and lower décalages) and the edgewise subdivision criterion, introduces free decomposition spaces arising from outer face complexes as examples, and assumes only basic knowledge of simplicial methods and category theory.","tokens_in":1651,"tokens_out":316,"duration_ms":9806,"significance":"The central equivalence is a standard result (Gálvez-Carrillo–Kock–Tonks; Dyckerhoff–Kapranov), so the paper's value is primarily expository: it unifies the two perspectives in one text, supplies concrete criteria and a source of examples via free decomposition spaces, and lowers the barrier to entry. This could usefully complement existing literature for readers already familiar with simplicial sets.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the active-inert factorization system 'provides a characterization equivalent to the 2-Segal condition,' but the introduction does not explicitly cite the original sources for this equivalence at the first mention; adding a pointer to Gálvez-Carrillo–Kock–Tonks or Dyckerhoff–Kapranov in §1 would help readers locate the primary references.","section":"Abstract / §1"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive report and recommendation to accept the manuscript. We appreciate the recognition of the paper's expository value in unifying the active-inert factorization perspective on decomposition spaces with the 2-Segal space viewpoint, along with the path space criterion, edgewise subdivision criterion, and examples from outer face complexes.","responses":[],"tokens_in":1100,"tokens_out":84,"duration_ms":11290,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this paper unifies the decomposition space and 2-Segal space perspectives by defining the former through the active-inert factorization system on the simplicial category. The equivalence is not new, but the paper collects the path space criterion, the edgewise subdivision criterion, and some examples in one place. It does well at keeping things accessible with minimal prerequisites and at generating examples from outer face complexes. Those free decomposition spaces seem like a practical contribution for anyone needing concrete instances. The limitation is that everything here is already known from prior papers. There are no new derivations or unexpected connections. The central claim holds up because the factorization system is standard and the pullback conditions line up with the 2-Segal maps, as the stress test notes. This is aimed at people working in simplicial category theory who need a clean overview. It is not for someone looking to cite a breakthrough. A serious referee should see it because good introductions help the field even if they do not advance the frontier. Recommendation: Yes, send it out for review as an expository piece.","headline":"This is a clear expository paper that unifies decomposition spaces via active-inert factorization with 2-Segal spaces, but adds no new theorems.","tokens_in":2089,"tokens_out":292,"would_cite":false,"duration_ms":18076,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"Definition 3.1 (Decomposition space). A simplicial space X is a decomposition space if it sends every active-inert pushout square to a pullback square."},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AlexanderDuality.lean","rs_theorem":"alexander_duality_circle_linking","paper_passage":"Lemma 2.2. The pair of subcategories (Δact, Δint) constitutes a factorization system on Δ."}],"headline":"Decomposition spaces via active-inert factorization on Δ are orthogonal to RS forcing chain","alignment":"orthogonal","rationale":"The paper's central machinery (active-inert factorization system on Δ, equivalence of decomposition spaces to 2-Segal spaces, path-space and edgewise-subdivision criteria) lives entirely in algebraic topology / higher category theory. RS framework derives spacetime, 3D, φ, J-cost, c=1, ℏ, G from a single distinction with no adjustable parameters; this paper contains none of those structures or theorems.","tokens_in":53630,"confidence":"high","tokens_out":302,"duration_ms":6429,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Decomposition spaces defined via active-inert factorization on simplices are equivalent to 2-Segal spaces.","keywords":["decomposition spaces","2-Segal spaces","simplicial sets","active-inert factorization","path space criterion","edgewise subdivision","outer face complexes"],"falsifier":"An explicit simplicial set that meets the 2-Segal condition but fails the active-inert decomposition axiom, or vice versa.","tokens_in":2454,"feed_emoji":"","tokens_out":428,"duration_ms":13196,"temperature":0.7,"pith_summary":"The paper shows that a simplicial object satisfies the decomposition space condition precisely when it satisfies the 2-Segal condition, by using the active-inert factorization system on the simplex category. This identification lets criteria such as the path space property for upper and lower décalages and the edgewise subdivision test apply interchangeably. The work also constructs free decomposition spaces from outer face complexes as a source of examples. A reader would care because both notions organize structures like homotopy associative algebras and higher categories, and the equivalence transfers constructions and proofs between the two viewpoints.","feed_headline":"Active-inert factorization equates decomposition spaces to 2-Segal spaces","feed_subtitle":"The equivalence transfers path-space and subdivision criteria and supplies examples from outer face complexes.","key_machinery":"The active-inert factorization system on the simplex category, which splits every map into an active part followed by an inert part and supplies the decomposition space axiom.","core_discovery":"A simplicial object is a decomposition space when every active map factors uniquely through an inert map in the active-inert factorization system of the simplex category; this condition is equivalent to the 2-Segal condition. The equivalence is proved directly, the path space criterion is derived from it, and the edgewise subdivision is shown to preserve the property.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Active-inert factorization unifies decomposition and 2-Segal spaces","Decomposition spaces match 2-Segal via active-inert factorization","Path space criterion characterizes decomposition spaces","Edgewise subdivision preserves decomposition space property","Free decomposition spaces from outer face complexes"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The active-inert factorization system on the simplex category exists and its induced condition matches the 2-Segal condition exactly.","fun_headline_variants_meta":{"raw":{"variants":["Active-inert factorization unifies decomposition and 2-Segal spaces","Decomposition spaces match 2-Segal via active-inert factorization","Path space criterion characterizes decomposition spaces","Edgewise subdivision preserves decomposition space property","Free decomposition spaces from outer face complexes"]},"model":"grok-4.3","cost_usd":0.005141,"raw_usage":{"total_tokens":2426,"prompt_tokens":524,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":51412000,"prompt_tokens_details":{"text_tokens":524,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1832,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":524,"tokens_out":70,"duration_ms":10206,"temperature":1.0,"reasoning_tokens":1832,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-23T21:24:48.535024+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit simplicial set that meets the 2-Segal condition but fails the active-inert decomposition axiom, or vice versa.","supporting_citations":[],"review_version":1}