{"id":"2083039a-196f-4180-a7d5-b062919ccf32","arxiv_id":"1908.04949","paper_version":4,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For compact Lie group actions on metrizable spaces, equivariant A-category is invariant under Morita equivalence, yielding Morita-invariant equivariant LS-category and invariant topological complexity, and hence an orbifold topological complexity.","lead":"The authors prove that two equivariant invariants, Lusternik-Schnirelmann category and invariant topological complexity, are unchanged under Morita equivalence of group actions. This makes the invariant topological complexity a well-defined numerical invariant for orbifolds, confirming a previously stated conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof is coherent and the cited decomposition Proposition 2.5 is standard.","rationale":"The reader's weakest_assumption correctly identifies Proposition 2.5 as the main external dependency. After careful review, I find the internal argument of the paper sound. Lemma 5.3 and Lemma 5.7 establish the quotient case, Lemma 5.8 establishes the induction case via two applications of the quotient case, and Theorem 5.1 follows by composing these pieces. The use of the equivariant covering homotopy property in Lemma 5.7 is legitimate because the quotient map X -> X/K is a numerable (G,α,K)-bundle under the stated hypotheses, and the lifting of a G/K-equivariant homotopy to a G-equivariant homotopy is exactly what the G-CHP provides. The passage from the equivariant A-category to invariant topological complexity in Corollary 5.10 is also sound, since the product of an essential equivalence is an essential equivalence and the saturation of the diagonal relation maps correctly. No internal inconsistency or overlooked gap was found. The unproved Proposition 2.5 is a standard cited theorem from Pronk-Scull, and the paper's scope of compact Lie groups acting on metrizable spaces is within the natural range of that theorem, so the accept verdict should stand unchanged.","tokens_in":11965,"tokens_out":22108,"duration_ms":222139,"concrete_test":"Verify Proposition 2.5 in the compact Lie/metrizable setting: for an essential equivalence ψ⋉ǫ: G⋉X -> H⋉Y, let K = ker ψ and check that K acts freely on X, and that the induced map G/K⋉X/K -> H⋉Y is an induction up to isomorphism by constructing the H-equivariant homeomorphism H×_{G/K}(X/K) ≅ Y using essential surjectivity and full faithfulness. If this homeomorphism is continuous with continuous inverse under the stated metrizability assumptions, the decomposition holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the decomposition of any essential equivalence into a quotient by a normal subgroup acting freely and an induction (Proposition 2.5), cited from Pronk-Scull and not proved in the paper. This is the most load-bearing external dependency: if the decomposition failed or required extra hypotheses not present for compact Lie actions on metrizable spaces, the proof of Theorem 5.1 would not go through. I checked the internal lemmas and found them sound: Lemma 5.7 correctly applies the G-covering homotopy property from [17] to lift categorical homotopies, and Lemma 5.8 correctly reduces the induction case to two applications of the quotient case via the free actions of G and H on G×X. The other cited input, Theorem 4.4 from [13], is also standard and consistent with the definitions. No internal gap, invalid step, or counterexample is apparent. The only point of fragility is the unproved Proposition 2.5, but it is a standard theorem in the stated setting, so this is not a significant objection.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the equivariant A-category of Clapp and Puppe is invariant under Morita equivalence for compact Lie group actions on metrizable spaces. The proof factorizes any essential equivalence of translation groupoids, via Proposition 2.5, into a quotient map by a normal subgroup acting freely and an induction map, and establishes invariance for each factor. The equivariant principal bundle techniques of [17] supply the needed covering homotopy property in the quotient step, while the induction step is reduced to two applications of the quotient case. Corollaries 5.9 and 5.10 then give Morita invariance of equivariant LS-category and of invariant topological complexity, and Section 6 uses this to define an orbifold invariant topological complexity.","tokens_in":12136,"tokens_out":17583,"duration_ms":158084,"significance":"If the result stands, it provides new Morita invariants for orbifolds, confirming a conjecture from [1] and extending the program of [19] to Lusternik-Schnirelmann category and topological complexity. The proof is carefully structured: Lemma 5.3 gives an explicit equivariant compression through the quotient, Lemma 5.7 carefully applies the equivariant covering homotopy property to lift categorical homotopies, and Lemma 5.8 cleanly reduces the induction case to two quotient cases. The paper also correctly notes that ordinary equivariant topological complexity is not a Morita invariant, so the distinction between the two equivariant notions is essential. The higher invariant topological complexities are treated as well, giving additional Morita invariants.","major_comments":[],"minor_comments":[{"comment":"The statement of Theorem 5.1 reads \"A cat_G(X) = A' cat_H(X)\", but the right-hand side should refer to the target space Y, i.e. \"A' cat_H(Y)\". The subsequent argument and corollaries use the intended equality with Y.","section":"Theorem 5.1"},{"comment":"The proof of Theorem 5.1 depends entirely on the factorization of essential equivalences into a quotient map and an induction map. Since the paper states that the argument of [19] works for topological groupoids, the authors should either provide a proof of this topological version or cite a precise theorem in the literature that covers translation groupoids of compact Lie group actions on metrizable spaces.","section":"Proposition 2.5"},{"comment":"The same notation \"TCG(X)\" is used in Definition 4.1 for equivariant topological complexity and in Definition 4.3 for invariant topological complexity. This is confusing because Corollaries 5.10 and 5.12 rely on the distinction; please use different notation, such as an underline or overline, consistently.","section":"Definitions 4.1 and 4.3"},{"comment":"In Corollary 5.10 the phrase \"the saturation ... of ǫ(ℸ_{G×G}(X))\" should read \"of (ǫ×ǫ)(ℸ_{G×G}(X))\", since ℸ_{G×G}(X) is a subset of X×X. The same correction applies in Corollary 5.12, where the map should be the n-fold product ǫ^n on X^n.","section":"Corollaries 5.10 and 5.12"},{"comment":"The notational remark \"if N is a normal subgroup of X, we write N ⊳ X\" should say \"of G\", not \"of X\".","section":"After Proposition 2.5"},{"comment":"The sentence \"Orbifolds were first introduced by Satake [21]\" cites [21], which is Schwarz's paper on the genus of a fiber space; a reference to Satake's original work should be added, or the citation should be corrected.","section":"Section 6"},{"comment":"In Definition 6.4 the same letter X is used for the orbifold and for the underlying space of the presentation groupoid G⋉X. This is potentially confusing; using different symbols (for example, O for the orbifold and X for the presentation space) would improve readability.","section":"Definition 6.4"},{"comment":"In the statement of Theorem 5.2, part (1) does not use the subgroup H that is introduced at the start of the sentence. The statement would be cleaner if H was introduced only where needed, or if the sentence structure was adjusted.","section":"Theorem 5.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound and a good fit for the journal. The only substantive request is to clarify the status of the topological version of Proposition 2.5, but this appears to be a standard result and does not undermine my confidence in the proof. The remaining issues are typographical and notational."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does what it says: it proves that Clapp-Puppe equivariant A-category, hence equivariant LS-category, and invariant topological complexity are Morita invariant for compact Lie actions on metrizable spaces, and it uses that to define an invariant topological complexity for representable orbifolds. The conjecture from Angel-Colman is settled. That is real progress.\n\nThe proof is well structured. Theorem 5.1 reduces to the two structural pieces from Pronk-Scull: quotient by a free normal subgroup, and induction. Lemma 5.3 handles one inequality directly; Lemma 5.7 uses the numerable equivariant principal bundle structure and the G-covering homotopy property from Murayama-Shimakawa to get the reverse. Lemma 5.8 is a neat double-quotient argument with G×H acting on G×X. I checked the continuity and equivariance checks; they are careful and correct. Corollaries 5.9, 5.10, and 5.12 follow cleanly.\n\nThe main external dependencies—the decomposition of essential equivalences (Prop 2.5) and the equivariant bundle homotopy property (Cor 5.6)—are standard and correctly cited. They are not proved in the paper, but they are not exotic; I would not hold that against the paper. The proof is not circular in any load-bearing way.\n\nSoft spots are minor. The statement of Theorem 5.1 has a typo: the right side should be A' cat_H(Y), not A' cat_H(X). I would have liked one explicit sentence noting that the orbifold invariant is defined only for representable orbifolds, though the paper does say that.\n\nI do not see a load-bearing flaw. This is a credible, useful paper for anyone working in equivariant LS-category, topological complexity, or orbifold invariants. It deserves a serious referee; I would send it out and expect acceptance after minor corrections.","headline":"A clean, correct proof that equivariant LS-category and invariant topological complexity are Morita invariant, settling a conjecture and giving orbifolds a topological complexity.","tokens_in":12682,"tokens_out":1814,"would_cite":true,"duration_ms":18029,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55M30","55P91","55R91"],"pacs":[],"model":"deepseek-v4-flash","headline":"Morita equivalence preserves equivariant LS-category and invariant topological complexity, giving orbifolds a topological complexity.","keywords":["equivariant LS-category","equivariant A-category","Morita equivalence","translation groupoids","invariant topological complexity","orbifolds","equivariant principal bundles","compact Lie group actions"],"falsifier":"Compute the invariant topological complexity of the free rotation actions of $\\mathbb{Z}_2$ and $\\mathbb{Z}_3$ on $S^1$, which are Morita equivalent with quotient $S^1$; the theorem predicts both equal $\\mathrm{TC}(S^1)$, so any calculation producing different values would settle the claim false.","tokens_in":11758,"feed_emoji":"🤖","tokens_out":9606,"duration_ms":90656,"temperature":0.7,"pith_summary":"The paper proves that Morita equivalent group actions have the same generalized equivariant LS-category, and therefore the same invariant topological complexity. Morita equivalence is the standard notion of sameness for orbifold presentations, so this makes both invariants well-defined on representable orbifolds. The proof works by decomposing any Morita equivalence into quotient maps and induction maps, then showing each step preserves the equivariant A-category using a homotopy property of equivariant principal bundles. The original equivariant topological complexity is shown not to be Morita invariant, so the paper singles out the invariant version as the right orbifold motion-planning invariant.","feed_headline":"Morita equivalence preserves equivariant category","feed_subtitle":"Equivariant LS-category and invariant motion-planning complexity are shown to be orbifold invariants.","key_machinery":"The object carrying the argument is the equivariant $\\mathcal{A}$-category of a $G$-space: the least number of invariant open sets in a cover whose inclusion maps deform up to $G$-homotopy into members of a prescribed class $\\mathcal{A}$ of invariant subsets. The proof machinery is the decomposition of any essential equivalence into two elementary maps—a quotient map by a freely acting normal subgroup and an induction map along a subgroup inclusion—together with the fact that numerable equivariant bundles with compact structure group satisfy a $G$-covering homotopy property, which lets categorical homotopies be lifted from a quotient $X/K$ back to $X$.","core_discovery":"The central result, Theorem 5.1, states that for an essential equivalence $\\psi\\ltimes\\epsilon\\colon G\\ltimes X\\to H\\ltimes Y$ between translation groupoids and any class $\\mathcal{A}$ of $G$-invariant subsets of $X$, one has $\\mathcal{A}\\,\\mathrm{cat}_G(X)=\\mathcal{A}'\\,\\mathrm{cat}_H(Y)$, where $\\mathcal{A}'$ is the saturation of $\\epsilon(A)$ under the $H$-action. The proof establishes this by proving that quotient maps $G\\ltimes X\\to G/K\\ltimes X/K$ and induction maps $H\\ltimes X\\to G\\ltimes(G\\times_H X)$ each preserve the equivariant $\\mathcal{A}$-category; the quotient step uses the $G$-covering homotopy property of numerable equivariant bundles, and the induction step follows by applying the quotient result twice. Consequently, equivariant LS-category and invariant topological complexity are invariants of Morita equivalence for compact Lie group actions on metrizable spaces, and the invariant topological complexity of a presentation defines an orbifold invariant (Definition 6.4).","pith_inferences":["An implicit consequence is that any equivariant homotopy invariant intended for orbifolds must be unchanged by free quotient maps; this explains why the standard equivariant topological complexity cannot work and suggests a general criterion for orbifold invariants.","Because the proof only needs the decomposition of essential equivalences and the covering homotopy property, the same method is likely to transfer other equivariant invariants—such as equivariant cohomology or K-theory—to orbifolds whenever the corresponding bundles satisfy the analogous lifting property.","The authors speculate that the result extends to proper actions of discrete groups on ANRs; verifying Theorem 5.1 in that setting would broaden orbifold topological complexity beyond compact Lie presentations."],"forward_implications":["Equivariant LS-category $\\mathrm{cat}_G(X)$ is a Morita invariant for compact Lie group actions on metrizable spaces, so it can be assigned to a representable orbifold independently of its presentation.","The invariant topological complexity $\\underline{\\mathrm{TC}}_G(X)$ is a Morita invariant, while the original equivariant topological complexity is not; the free $S^1$-action on $S^1$ versus the trivial action on a point shows the difference.","Definition 6.4 gives an orbifold topological complexity $\\mathrm{TC}_{\\mathcal{O}}(\\mathcal{X})$ by evaluating the invariant topological complexity on any translation-groupoid presentation.","The higher invariant topological complexities $\\underline{\\mathrm{TC}}_{G,n}(X)$ are also Morita invariant by the same argument.","New lower bounds such as $\\max_{K\\triangleleft G}\\mathrm{cat}((X/K)^{G/K})\\le \\mathrm{cat}_G(X)$ follow, generalizing known inequalities."],"supporting_citations":[{"why":"Supplies the decomposition of every essential equivalence into quotient and induction maps (Proposition 2.5), the structural starting point of the proof.","marker":"[19]"},{"why":"Provides the G-covering homotopy property for numerable equivariant bundles (Corollary 5.6) used to lift categorical homotopies from quotients.","marker":"[17]"},{"why":"Defines the generalized A-category whose equivariant version is the invariant under study.","marker":"[5]"},{"why":"Introduces the invariant topological complexity and its characterization via the saturated diagonal, which Corollary 5.10 relies on.","marker":"[13]"},{"why":"Defines the original equivariant topological complexity and is used to show it is not Morita invariant.","marker":"[7]"},{"why":"Supplies the principal K-bundle statement used in Proposition 5.4 for free actions of a closed normal subgroup.","marker":"[4]"},{"why":"Provides the theory of (Gamma,alpha,G)-bundles, including local triviality and the category equivalence used in Lemma 5.8.","marker":"[9]"},{"why":"Sets up orbifolds as Morita equivalence classes of proper foliation groupoids, which Definition 6.4 depends on.","marker":"[16]"}],"fun_headline_variants":["Equivariant LS-category survives Morita equivalence","Orbifold invariant from equivariant motion planning","Morita invariance for equivariant category and complexity","Invariant topological complexity defines orbifold invariant","Equivariant LS-category is Morita invariant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every Morita equivalence can be broken into two simple kinds of maps—dividing out a freely acting normal subgroup and inducing a subgroup action—and the paper cites rather than proves this decomposition; if it fails in the setting of compact Lie groups, the main theorem has no proof.","fun_headline_variants_meta":{"raw":{"variants":["Equivariant LS-category survives Morita equivalence","Orbifold invariant from equivariant motion planning","Morita invariance for equivariant category and complexity","Invariant topological complexity defines orbifold invariant","Equivariant LS-category is Morita invariant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000161,"raw_usage":{"total_tokens":1186,"prompt_tokens":843,"completion_tokens":343,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":269}},"tokens_in":459,"tokens_out":343,"duration_ms":2937,"temperature":1.0,"reasoning_tokens":269,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:28:14.728437+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the invariant topological complexity of the free rotation actions of $\\mathbb{Z}_2$ and $\\mathbb{Z}_3$ on $S^1$, which are Morita equivalent with quotient $S^1$; the theorem predicts both equal $\\mathrm{TC}(S^1)$, so any calculation producing different values would settle the claim false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the decomposition of every essential equivalence into quotient and induction maps (Proposition 2.5), the structural starting point of the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the G-covering homotopy property for numerable equivariant bundles (Corollary 5.6) used to lift categorical homotopies from quotients."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the generalized A-category whose equivariant version is the invariant under study."},{"cited_title":"Lon- don Math","cited_arxiv_id":null,"evidence_quote":"Introduces the invariant topological complexity and its characterization via the saturated diagonal, which Corollary 5.10 relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the original equivariant topological complexity and is used to show it is not Morita invariant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the principal K-bundle statement used in Proposition 5.4 for free actions of a closed normal subgroup."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the theory of (Gamma,alpha,G)-bundles, including local triviality and the category equivalence used in Lemma 5.8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets up orbifolds as Morita equivalence classes of proper foliation groupoids, which Definition 6.4 depends on."}],"review_version":1}