{"id":"96618781-c078-4fd5-bc04-718858b9bb67","arxiv_id":"2509.07285","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A general sufficient condition, 'goodness', under which cohomology of cosimplicial sheaves of modules over Lie groupoids is Morita invariant, unifying earlier case-by-case proofs.","lead":"This math paper gives a unified way to prove that cohomology theories associated to Lie groupoids are unchanged under Morita equivalence, using sheaves of modules on the groupoid's nerve. The framework packages several previously separate invariance proofs into two general theorems and recovers known results as corollaries.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.12's left augmentation is not proved: the 'symmetrical' right-retraction is never defined, and with the stated total-differential sign the naive map e -> φ_0^{n+1}e is not a chain map into Tot(Ch(V)_shift).","rationale":"The reader's verdict of CONDITIONAL is reasonable: the paper gives a plausible unified framework and many explicit constructions, but the written proof of a central quasi-isomorphism is incomplete. I agree with the reader that 'goodness' is under-characterized and that Lemma 8.11's module lift is not fully verified. However, the single most load-bearing concern is Lemma 4.12's left augmentation: every main theorem uses both L_V and B_V, and the proof of L_V is explicitly deferred to an undefined 'right retraction.' Moreover, a direct computation with the stated total differential and Definition 4.11 shows a sign obstruction to the naive chain map; the paper does not address this. This is a proof gap rather than a demonstrated counterexample, so I do not recommend rejection. The existing CONDITIONAL verdict already asks for the missing verifications to be supplied; my analysis identifies which verification is most critical. The concrete test on the constant cosimplicial vector space would settle whether the sign obstruction is real or whether a simple sign correction restores the argument.","tokens_in":29241,"tokens_out":56137,"duration_ms":644336,"concrete_test":"Take the constant cosimplicial vector space V(n)=R with all coface and codegeneracy maps the identity. Compute L_V(e) for e in V^1, the simplicial differential δ^1 (which is 1), and the total differential of Ch(V)_shift at D^{1,0} using the paper's Definitions 2.20 and 3.7. One obtains d_tot L(e) = -L(δ e), so the stated L_V is not a chain map into the total complex. Then attempt to write down the promised 'right retraction' for this example; if no sign/retraction correction is provided, Lemma 4.12 is unproved and the main diagram lacks a leg.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sections 8 and 9 (Theorems 8.3, 8.7, 8.9, 9.9, 9.11) all rely on the diagram G -> G_shift, whose cochain-level left and bottom augmentations L_V and B_V must both be quasi-isomorphisms. Lemma 4.12 proves B_V via the standard column retraction, but for L_V it only says 'a symmetrical argument can be applied ... one must use right retractions ... which we have not formally defined or constructed.' That is a missing proof of a load-bearing leg. The gap is not merely expository. With Definition 2.20's total differential d_tot = d_H + (-1)^p d_V, and Definition 4.11's L_V(e) = φ_0^{n+1}e in D^{n,0}=V(n+1), the cosimplicial identity (CS1) gives d_H L_V=0 and d_V L_V e = L_V δ e, but d_tot L_V e = (-1)^n L_V δ e. Thus the naive map is not a chain map into the total complex; a sign correction is needed before the augmentation lemma can be applied. The paper supplies neither the correction nor the promised right-retraction. Since Theorems 8.7 and 9.11 are the paper's central claims, this unverified quasi-isomorphism is the most load-bearing point. The 'goodness' issue is real but secondary: Lemma 8.11's lift is plausible when the index is r_{n+m}, but the left-leg gap affects every application.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a unified framework for proving Morita invariance of cohomology theories associated to Lie groupoids. The central objects are cosimplicial sheaves of modules (or complexes) on the nerve of a Lie groupoid, subject to a 'goodness' condition (Definition 7.8). The main results, Theorem 8.9 and Theorem 9.9, assert that for a good cosimplicial module (or complex) E over G and a weak equivalence F:H→G, the natural pullback e↦e⊗1 induces an isomorphism in cohomology. Parallel statements for sheaves on the big site are given as Theorems 8.7 and 9.11. The proofs use the shift double (décalage) construction, the associated double/triple complexes, partitions of unity to patch local retractions, and an F-double groupoid attached to a groupoid homomorphism. Known Morita-invariance results for groupoid cohomology, representation-valued cohomology, and Bott–Shulman cohomology are derived as corollaries.","tokens_in":29701,"tokens_out":9599,"duration_ms":112793,"significance":"If the main theorems are correct, the paper gives a single sufficient condition covering several formerly separate Morita-invariance results, and it offers a low-tech, explicit route via retractions and partitions of unity. The explicit construction of local retractions and their patching is a genuine strength. However, the validity of the framework currently depends on a missing proof of one leg of Lemma 4.12 and on an unproved claim that big-site sheaves are automatically good. These issues are local and likely repairable, but they are load-bearing for the stated theorems.","major_comments":[{"comment":"The left augmentation L_V is not proved to be a quasi-isomorphism, and as stated it is not a chain map with respect to the paper's total differential. Definition 4.11 sets L_V(e)=φ_0^{n+1}e in D^{n,0}. In the total complex (Definition 2.20), d_tot = d_H + (-1)^p d_V, so on D^{n,0} the vertical component is multiplied by (-1)^n. Using (CS1) one gets d_H L_V=0 and d_V L_V = L_V δ, hence d_tot L_V = (-1)^n L_V δ, not L_V δ. Thus L_V is not a morphism of cochain complexes without a supplemental Koszul sign. The proof of Lemma 4.12 also explicitly relies on 'right retractions' that are never defined or constructed. Since L is used in the commutative diagram in the proofs of Theorems 8.3, 8.7, 8.9, 9.9, and 9.11, this gap affects all central claims. Please supply the corrected sign and prove the required row acyclicity.","section":"§4.5, Lemma 4.12"},{"comment":"The 'goodness' condition is a strong, paper-specific hypothesis, and the paper asserts without proof that 'any module which is sufficiently functorial is easily seen to be good' and that any big-site sheaf is automatically good. These assertions underpin Theorems 8.7 and 9.11, as well as the application of Theorem 8.9 to Bott–Shulman cohomology. Since no characterization or proof is given, the theorems are conditional on an unverified property. Please either prove the claim for big-site sheaves and for the examples in Section 7, or explicitly state goodness as a hypothesis in the main theorems and restrict the corollaries accordingly.","section":"Definition 7.8 and §8.3"},{"comment":"Lemmas 2.22 and 9.3 are central to the argument and are stated without proof. Lemma 2.22 is standard and may be acceptable, but Lemma 9.3, the triple-complex augmentation lemma, is not standard in the same way and is only justified by a remark that it can be proved by collapsing directions. Since Theorem 9.9 and Theorem 9.11 depend on Lemma 9.3, please include a proof or a precise reference; at minimum, spell out the collapsing argument.","section":"§9.1, Lemma 9.3 and §2.4, Lemma 2.22"}],"minor_comments":[{"comment":"The formula for δL[rks]_n contains 'V(n+m)' where m is not defined. Probably it should be V(n+k) or the displayed index should be clarified.","section":"§3.3, Definition 3.6"},{"comment":"In Theorem 1.2, 'the pullback cosimplicial complex on G' should be 'on H' to match the statement in Section 9.3. There is also a typo 'Bott-Shulmann' in the Introduction.","section":"§1, Corollary 1.2 statement"},{"comment":"The sentence 'Since B_E⊗P_F^* is injective it must be a quasi-isomorphism' is not by itself a valid implication. The preceding claim that the map admits a left retract is the relevant reason; please rephrase to avoid a logical gap.","section":"§8.4, Lemma 8.10"},{"comment":"The final displayed line defines F^#: Ch(E)→Ch(F^*W); the target should be Ch(F^*E).","section":"§7.4, Definition 7.14"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the missing and partially incorrect proof of the left augmentation in Lemma 4.12. The issue appears fixable by adding the appropriate sign factor and constructing the promised right retractions, so I recommend major revision rather than rejection. Please also ask the authors to clarify the status of 'goodness' as an automatic property of big-site sheaves; this is a broader condition than the individual examples and should not be asserted without proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth reading, worth refereeing, but not ready as is. The core idea is real: package groupoid cohomology theories as sheaves of modules/complexes on the nerve, then reduce Morita invariance to a \"goodness\" lifting condition plus explicit retractions. The paper recovers Crainic, Arias Abad–Crainic, and Bott–Schulman cohomology as corollaries, which is a useful organizational contribution. The decalage/shift double setup is explicit, and the local-to-global partition-of-unity argument in Section 8 is sound in spirit.\n\nThe soft spot is load-bearing. Lemma 4.12 asserts that both the bottom and left augmentations into Ch(V)_shift are quasi-isomorphisms. The bottom argument works via the standard column retraction. For the left augmentation, the proof says a symmetrical argument can be applied and admits that the needed right retractions were never defined. That alone is a missing proof. Worse, with the stated total differential d_tot = d_H + (-1)^p d_V, the stated map L_V(e) = φ_0^{n+1} e in D^{n,0} satisfies d_tot L_V(e) = (-1)^n L_V(δ e), not L_V(δ e). So it is not a chain map as written; either the sign in the total differential or the definition of L_V needs adjustment. The paper supplies neither. Since Theorems 8.3, 8.7, 8.9, 9.9, and 9.11 all route through this map, the central claims are unproved in the current version.\n\nThe \"goodness\" hypothesis is a secondary concern. It is honest that the authors know no non-good example, and the condition is a domain assumption rather than a fitted conclusion, but it is still under-characterized. Lemma 2.22 and Lemma 9.3 are stated without proof; those look like standard/plausible facts and are not my main worry.\n\nWho gets value from this paper: anyone working on Lie groupoid cohomology, differentiable stacks, or Morita invariance of cohomological invariants. The framework is promising enough that I would send it to a serious referee rather than desk reject, with instructions that the sign issue and the missing retraction for Lemma 4.12 must be fixed. I would not cite the main theorems as established until that happens.","headline":"A genuinely useful unified framework for Morita invariance, but Lemma 4.12 is not proved and the stated left augmentation is not even a chain map with the paper's sign convention, so the main theorems do not yet follow as written.","tokens_in":30081,"tokens_out":4432,"would_cite":false,"duration_ms":52241,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22A22","58H05","18G30","55N30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that Morita invariance of groupoid cohomology, cohomology with coefficients in representations, and Bott–Schulman cohomology all follow from one condition on sheaves of modules over a groupoid's nerve.","keywords":["Lie groupoids","Morita equivalence","groupoid cohomology","cosimplicial modules","simplicial sheaves","Bott–Schulman cohomology","decalage","weak equivalence"],"falsifier":"Construct a Lie groupoid G and a naturally occurring cosimplicial module E over G, together with a local section r_0:U→G^1 of the target map, such that the associated target family {r_n} cannot be lifted to a family of comorphisms (r_n)^# satisfying (r_n)^#∘ϕ^{n+1}_i = Id for i=0 and (r_n)^#∘ϕ^{n+1}_i = ϕ^n_{i−1}∘(r_{n−1})^# for 1≤i≤n. Such an E would show goodness is not automatic and would remove that example from the coverage of Theorem 9.9.","tokens_in":29162,"feed_emoji":"📐","tokens_out":10989,"duration_ms":106365,"temperature":0.7,"pith_summary":"Lie groupoids are atlases for singular quotient spaces, and a cohomology theory built from a groupoid is only meaningful if it does not depend on which atlas one chooses—that is, if it is Morita invariant. This paper claims that a single framework proves Morita invariance for the standard theories: it views each as a sheaf of modules (or a sheaf of complexes) on the nerve of the groupoid, and shows that weak equivalences induce isomorphisms in cohomology whenever the sheaf is 'good' or comes from a sheaf on the big site. The proof is deliberately low-tech, based on the shift double of the nerve, an auxiliary double groupoid attached to the homomorphism, and local retractions patched together with partitions of unity. If correct, the paper replaces several separate Morita-invariance arguments with one sufficient condition.","feed_headline":"One sheaf criterion proves Morita invariance of groupoid cohomology","feed_subtitle":"The same cosimplicial-module argument covers plain cohomology, representation coefficients, and Bott–Schulman.","key_machinery":"The shift double (decalage) of a cosimplicial object, which arranges left and right shifts into rows and columns; the F-double groupoid attached to a homomorphism F:H→G; target families and the goodness condition; and left retractions of cosimplicial modules, assembled globally from local ones via partitions of unity. The proof factors the pullback through the shift double and shows the relevant augmentations are acyclic by constructing explicit left retracts.","core_discovery":"The central claim is Theorem 9.9 (big-site variant Theorem 9.11): for a weak equivalence F:H→G and a good cosimplicial complex E over G, the pullback F^#:E→F^*E, e↦e⊗1, is an isomorphism in cohomology; for a sheaf of complexes on the big site, E_F:E_H→E_G is an isomorphism. Goodness (Definition 7.8) is a Kan-like condition: every local section of the target map lifts to a compatible family of module comorphisms. The authors know no non-good example and consider sufficiently functorial modules good, but give no characterization. The corollaries—groupoid cohomology, representation coefficients, Bott–Schulman cohomology, and compactly supported versions—are instances of this single result.","pith_inferences":["If goodness turns out to be automatic for all cosimplicial complexes arising from natural bundles and geometric structures, then Theorem 9.9 would cover essentially every cohomology theory of this type, making the condition a convenience rather than a restriction.","The local-section-plus-partition-of-unity method suggests the same strategy should extend to simplicial manifolds satisfying mild Kan-like conditions, beyond Lie groupoids, as the authors themselves note.","A direct testable extension is to apply the framework to only partially functorial sheaves—compactly supported forms or distribution-valued sections—to see where goodness can fail.","The framework likely also gives Morita invariance for twisted de Rham cohomology associated to flat connections or local systems, whenever the twist is functorial enough to be good."],"forward_implications":["Morita equivalent Lie groupoids have isomorphic groupoid cohomology.","Cohomology with coefficients in a groupoid representation is Morita invariant, recovering earlier results without a separate argument.","Bott–Schulman cohomology of a Lie groupoid is Morita invariant.","Compactly supported versions of these cohomology theories are Morita invariant as well (Corollary 1.6).","Any sheaf of complexes on the big site of smooth manifolds—for example differential forms—produces a Morita-invariant cohomology theory."],"supporting_citations":[{"why":"supplies the definition and characterization of Morita equivalence via weak equivalences that the paper uses as its starting point.","marker":"[BX11]"},{"why":"source of the decalage/shift construction that organizes the double and triple complexes in the proof.","marker":"[Ill71]"},{"why":"earliest proof of Morita invariance of groupoid cohomology, the base case the new framework recovers and extends.","marker":"[Cra03]"},{"why":"earlier proof of Morita invariance for cohomology with coefficients in representations up to homotopy, recovered as a corollary.","marker":"[AC13]"}],"fun_headline_variants":["One sheaf criterion proves Morita invariance across groupoid cohomology theories","Kan-like 'good' condition ties together groupoid cohomology proofs","Theorem 9.9: Unified proof of Morita invariance for sheaves of modules","No known counterexample: 'good' modules make cohomology Morita invariant","Single proof covers Bott-Schulman and representation coefficients via sheaves"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The pullback theorem for cosimplicial complexes assumes the 'goodness' condition (Definition 7.8), and the paper gives no proof or characterization that natural examples satisfy it; if a natural example fails, the proof's local retraction step collapses and the Morita-invariance conclusion for that example does not follow.","fun_headline_variants_meta":{"raw":{"variants":["One sheaf criterion proves Morita invariance across groupoid cohomology theories","Kan-like 'good' condition ties together groupoid cohomology proofs","Theorem 9.9: Unified proof of Morita invariance for sheaves of modules","No known counterexample: 'good' modules make cohomology Morita invariant","Single proof covers Bott-Schulman and representation coefficients via sheaves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000617,"raw_usage":{"total_tokens":2634,"prompt_tokens":613,"completion_tokens":2021,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":357,"completion_tokens_details":{"reasoning_tokens":1921}},"tokens_in":357,"tokens_out":2021,"duration_ms":15614,"temperature":1.0,"reasoning_tokens":1921,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T22:28:18.945344+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a Lie groupoid G and a naturally occurring cosimplicial module E over G, together with a local section r_0:U→G^1 of the target map, such that the associated target family {r_n} cannot be lifted to a family of comorphisms (r_n)^# satisfying (r_n)^#∘ϕ^{n+1}_i = Id for i=0 and (r_n)^#∘ϕ^{n+1}_i = ϕ^n_{i−1}∘(r_{n−1})^# for 1≤i≤n. Such an E would show goodness is not automatic and would remove that example from the coverage of Theorem 9.9.","supporting_citations":[],"review_version":1}