{"id":"e2f644f7-28ff-4b18-9695-69f2da15afde","arxiv_id":"2607.17362","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Derived Lie n-groupoids carry shifted symplectic and lagrangian structures whose composition is well defined under transversality, yielding a unified derived symplectic reduction at critical values.","lead":"This paper introduces 'derived Lie n-groupoids' — smooth spaces with extra directions that make intersections and quotients behave well — and defines shifted symplectic structures on them. It proves a composition theorem for lagrangian correspondences under transversality and uses it to unify symplectic reduction procedures at critical moment-map values, including Hamiltonian, quasi-Hamiltonian, Poisson-Lie, and proper-groupoid cases.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quotient's status as 'the homotopy pullback' is not well-defined: Rem. 4.26 concedes independence from the choice of fibrant replacement is unproved, and Thm 7.5 builds on that choice.","rationale":"The paper is a substantial, careful development, and I find no clear algebraic error in the main reduction construction. The reader's verdict of CONDITIONAL is appropriate. I agree with the reader that weak symplectic linearizability is a load-bearing hypothesis and is verified only in the listed examples; §7's opening text itself admits the hypothesis has not been eliminated. However, I would locate the most serious unresolved point slightly differently: Theorem 7.5's statement that the quotient is the homotopy pullback is not yet a well-defined claim because §4.5's homotopy pullback depends on a chosen fibrant replacement, and Rem. 4.26 explicitly concedes that independence is not established. Since the theorem's constructed quotient is built from the chosen linearization data, the absence of an independence proof means the central claim linking the quotient to the derived-geometric homotopy pullback is under-supported. This does not destroy the paper's contribution — the construction still yields a 0-shifted symplectic quasi-smooth groupoid for each admissible choice — but it does justify keeping the verdict at CONDITIONAL rather than ACCEPT. A concrete comparison of two different connections/linearizations, as described in the test, would settle whether this is merely a missing proof or an actual failure of canonicity.","tokens_in":71566,"tokens_out":32208,"duration_ms":336703,"concrete_test":"In the proper groupoid setting of §7.3, fix a hamiltonian Γ-space M and an orbit O. Compute the quotient twice using two different principal connections θ in Thm 7.6 (equivalently, two different tubular neighbourhood embeddings Φ0). Then check whether the two resulting 0-shifted symplectic derived Lie groupoids are connected by a zigzag of strict symplectic equivalences (Def. 5.4). If they are not, M//_OΓ depends on the choice of replacement and Thm 7.5(1)'s homotopy-pullback assertion is not justified as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 7.5(1) asserts that M//_O Γ is the homotopy pullback of the two 1-shifted lagrangians µ• and i•. But §4.5 defines a homotopy pullback via a chosen fibrant replacement ТН, and Rem. 4.26 explicitly states that the paper does not demonstrate independence of the choice of fibrant replacement, nor does it show the expected 2-morphism compatibility between two possible replacements. In Thm 7.5 the replacement is U•, built from chosen linearization data (Φ•, Ω_lin, η). Different linearizations, tubular neighbourhood embeddings, or connections (e.g. the connection θ in Thm 7.6) will generally produce different U•. No argument in the paper shows the resulting quotients are connected by symplectic Morita equivalences. Thus the central assertion \"is the homotopy pullback\" — the part that makes the construction canonical and connects it to the derived-geometric picture — is not yet a well-defined invariant. This is more internal than the separate limitation, which §7.7.1? actually §7 intro concedes, that weak symplectic linearizability is only verified in examples and not established as a general hypothesis. The theorem may be read as an existence statement relative to chosen data, but then the phrase \"the symplectic quotient\" and \"the homotopy pullback\" overstates uniqueness.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a differential-geometric model of derived higher geometry. It introduces derived Lie n-groupoids as n-groupoid objects in the category of derived manifolds equipped with the pretopology of locally split fibrations, and proves (Thm. 4.18) that these form an incomplete category of fibrant objects with stalkwise weak equivalences, Kan fibrations, and hypercovers as acyclic fibrations. It then defines shifted differential forms via a simplicial graded de Rham triple complex, introduces IM-pairings through an adjusted Eilenberg–Zilber map, and uses these to define m-shifted symplectic structures and m-shifted lagrangian morphisms. The main structural result is Thm. 6.10, a composition theorem for shifted lagrangian correspondences under levelwise transversality and a mild groupoid condition. The final part applies this machinery to symplectic reduction at critical values: for a 1-shifted symplectic Lie 1-groupoid (quasi-symplectic groupoid) and a hamiltonian space, Thm. 7.5 constructs, under a weak symplectic linearizability hypothesis, a quasi-smooth 0-shifted symplectic derived Lie groupoid presented as the homotopy pullback of the moment and orbit lagrangians. Examples include proper groupoids, quasi-Hamiltonian reduction at the unit, and Lu reduction for Poisson–Lie group actions.","tokens_in":71917,"tokens_out":14696,"duration_ms":153511,"significance":"If the main claims hold, the paper provides a substantial extension of shifted symplectic geometry from algebraic geometry and higher Lie groupoids to a concrete derived differential-geometric setting. The explicit iCFO construction, the detailed proof of the lagrangian composition theorem, and the unified treatment of several reduction procedures at critical values are genuine contributions. The paper is also commendably explicit about its hypotheses and its open points, including Rem. 4.26 on the non-canonicality of homotopy pullbacks and the acknowledgment in the introduction and §7 that weak symplectic linearizability is not established in general. The composition theorem (Thm. 6.10) is a strong, internally well-developed result that does not depend on the problematic homotopy-pullback formalism. The reduction theorem is valuable as a conditional construction, but its formulation as producing 'the' homotopy pullback currently overstates its invariance properties.","major_comments":[{"comment":"The homotopy pullback used in Thm. 7.5(1) is not shown to be independent of the chosen fibrant replacement. Definition 4.25 defines it via a chosen replacement eH•, and Rem. 4.26 explicitly states that independence from this choice and the expected 2-morphism compatibility are not demonstrated. In Thm. 7.5 the replacement is U•, constructed from the linearization data (Φ•, Ω_lin, η) of Def. 7.3; different linearizations, tubular neighbourhood embeddings, or the connection θ appearing in Thm. 7.6 will generally produce different U•. No argument is given that the resulting quotients are connected by symplectic Morita equivalences or by any canonical comparison map. Thus the assertion that the quotient 'is the homotopy pullback' is not yet a well-defined invariant. At present the theorem is best read as an existence statement relative to chosen data. Please either prove independence/canonic","section":"§4.5, Rem. 4.26; Thm. 7.5(1)"},{"comment":"The central reduction theorem is conditional on weak symplectic linearizability at the orbit O, a property that is verified in the examples of §§7.3–7.6 but is not established for general quasi-symplectic groupoids. The authors are transparent about this in the introduction and the beginning of §7, and the theorem is stated as a conditional result. Nevertheless, the advertised scope — a unified framework for reduction at critical values — is weaker than a general singular-reduction theorem. This should be reflected more prominently in the abstract and Theorem 1.3, so that readers do not take Thm. 7.5 as a general theorem about arbitrary quasi-symplectic groupoids.","section":"§7 intro and Def. 7.3; Thm. 7.5"},{"comment":"The remark that the composition theorem recovers the homotopy-pullback picture of [PTVV13] requires, in the notation of Rem. 6.13, the existence of an (m−1)-shifted form β̃′• and an (m−2)-shifted form γ′• satisfying β′• − σ^*β̃′• = Dγ′•. The text says this is 'expected to be automatic', but no proof is supplied. Since this is the bridge between the paper's strictly transversal composition theorem and the general derived-geometric homotopy-pullback composition, the statement in Rem. 6.13 is conditional. This does not affect Thm. 6.10 itself, but the conditional nature should be marked more clearly, and the expected result should either be proved or labelled as a conjecture.","section":"§6.2, Rem. 6.13"}],"minor_comments":[{"comment":"Typo: 'sympelctic' should be 'symplectic'.","section":"Thm. 6.11"},{"comment":"The text calls η• a '1-shifted 2-form', but by (73) and Def. 6.2 it is a 0-shifted 2-form (since Γ• is 1-shifted symplectic). Please correct the terminology.","section":"Prop. 7.4(2)"},{"comment":"Typo: 'genenral' should be 'general'.","section":"Rem. 4.26"},{"comment":"The notation U• is used both for an open subgroupoid of the normal bundle ν• and for the derived Lie subgroupoid obtained from (71). This is a frequent source of confusion; consider distinguishing the two, for example by a separate script or by writing (U•, 0) for the derived object.","section":"§7.2.3, Prop. 7.4"},{"comment":"The BFV/BRS discussion is presented as a 'suspect[ion]' rather than a theorem. That is fine, but it may be clearer to place it explicitly in an outlook paragraph, separated from the established results.","section":"§7.4.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is very long and technically rich. The main issue is not the internal correctness of the iCFO construction or the lagrangian composition theorem, but the canonicality of the homotopy pullback in Thm. 7.5. The authors themselves concede in Rem. 4.26 that independence is unproved; this undercuts the strong wording of Thm. 7.5(1). A revision that either proves compatibility between different fibrant replacements or carefully reformulates the theorem as an existence result relative to chosen linearization data would make the paper acceptable. I would also suggest that the abstract and Theorem 1.3 be reworded to make the linearizability hypothesis unavoidable, rather than appearing as a mild technical condition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real paper with real new content, and it deserves a serious referee. But read Thm 7.5 with caveats: the 'homotopy pullback' claim is weaker than it looks, and the linearization hypothesis is only checked in examples.\n\nWhat's actually new: derived Lie n-groupoids over the locally split pretopology on derived manifolds, an iCFO structure (Thm 4.18) that recovers the expected homotopy theory, a working definition of shifted symplectic/lagrangian structures on these objects, a composition theorem for lagrangian correspondences under transversality (Thm 6.10), and a reduction theorem at critical values that unifies ordinary Hamiltonian, quasi-Hamiltonian, Poisson-Lie, and Mikami-Weinstein reduction. The proofs of the iCFO and composition theorems are detailed and look correct under the stated hypotheses. The discussion of the [RZ20] axioms (Rem. 2.1) is careful and honest.\n\nWhere it gets soft. First, the reduction theorem (Thm 7.5) depends on weak symplectic linearizability (Def. 7.3), which the paper verifies for proper groupoids, quasi-Hamiltonian units, and coboundary Poisson-Lie cases, but not in general. The authors say as much in the preamble to §7, so this is an acknowledged limitation rather than a hidden one.\n\nSecond, and more concerning, the phrase 'is the homotopy pullback' in Thm 7.5(1) overstates what is proven. Rem. 4.26 explicitly concedes the paper does not show independence from the choice of fibrant replacement. In Thm 7.5 the replacement U• is built from chosen linearization data (Φ•, Ω_lin, η); different choices will generally produce different U•, and no argument connects the resulting quotients by symplectic Morita equivalences. So the construction is well-defined relative to chosen data, but calling it 'the' symplectic quotient or 'the' homotopy pullback goes beyond the proof. This is not a fatal mathematical error — the theorem still produces a 0-shifted symplectic derived groupoid for each choice of data — but it is a substantial gap between the statement and the advertised canonicity.\n\nMinor soft spots: some results in the epilogue (Prop. 7.12, Thm 7.14) are only sketched, and several auxiliary statements rely on future work (e.g., Morita invariance of the de Rham complex). The citation pattern looks fine; the heavy reliance on [RZ20], [CZ23], [BLX24] etc. is legitimate, and the overlap with [RZ20] involves published independent results.\n\nBottom line: this paper is for symplectic and derived geometers who want a differential-geometric model for shifted symplectic structures. It deserves a serious referee, though the referee should push on the canonicity of the homotopy pullback and on whether the linearization hypothesis can be weakened. I'd take it to a reading group and would cite it if I were working in the area.","headline":"Solid new framework for shifted symplectic derived higher groupoids, but Thm 7.5's 'homotopy pullback' claim overreaches the proof.","tokens_in":72363,"tokens_out":3489,"would_cite":true,"duration_ms":32015,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D17","53D20","58A50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper develops derived Lie n-groupoids with shifted symplectic structures and proves that, under a weak symplectic linearizability assumption, singular symplectic reduction is a quasi-smooth 0-shifted symplectic derived Lie groupoid, r","keywords":["derived Lie n-groupoids","shifted symplectic structures","lagrangian correspondences","symplectic reduction","singular reduction","moment maps","quasi-smooth derived manifolds"],"falsifier":"Look for a 1-shifted symplectic Lie groupoid and an orbit whose normal-bundle groupoid admits no weak symplectic linearization: concretely, check whether the pullback of the 2-form can be written as a linear part plus a simplicial coboundary with the prescribed restriction to the orbit. A single orbit where this fails shows the theorem does not apply; alternatively, compute the tangent-complex pairing at a classical point of the derived quotient in a critical example and check whether the map to the shifted cotangent complex is a quasi-isomorphism.","tokens_in":71472,"feed_emoji":"🔄","tokens_out":7034,"duration_ms":73271,"temperature":0.7,"pith_summary":"The paper's central aim is to build a smooth, differential-geometric version of shifted symplectic geometry in which ordinary symplectic reduction remains meaningful even when the moment map has critical values. It introduces derived Lie n-groupoids — simplicial objects in a category of derived manifolds — and defines shifted symplectic forms and lagrangian morphisms on them. Its main structural result is a composition theorem for lagrangian correspondences under transversality. Its main application states that, when a quasi-symplectic groupoid is weakly symplectically linearizable at an orbit, the symplectic quotient is a quasi-smooth 0-shifted symplectic derived Lie groupoid, equal to the homotopy pullback of the moment lagrangian and the orbit lagrangian. If correct, singular symplectic reduction acquires a single derived object on which the reduced form is nondegenerate, with classical reduced spaces appearing as the classical locus.","feed_headline":"Singular symplectic reduction becomes a derived Lie groupoid","feed_subtitle":"Under a linearizability condition, critical moment-map levels get a nondegenerate reduced symplectic form.","key_machinery":"Derived Lie n-groupoids: simplicial objects in derived manifolds — non-positively graded manifolds with a cohomological vector field — satisfying horn-filling conditions with respect to the pretopology of locally split fibrations. Shifted symplectic forms are closed shifted 2-forms whose IM-pairing, built through an adjusted simplicial shuffle map, induces a quasi-isomorphism from the tangent complex to the shifted cotangent complex at classical points. The load-bearing construction for reduction is the replacement of the orbit subgroupoid by the derived normal-bundle groupoid; weak symplectic linearization supplies compatibility data that make this replacement a lagrangian fibration, so the","core_discovery":"At the center is the assertion that the failure of transversality in symplectic reduction can be repaired by derived geometry. For a 1-shifted symplectic Lie groupoid, a hamiltonian space, and an orbit, the orbit inclusion is replaced by a derived groupoid built from the normal bundle of the orbit; a weak symplectic linearizability condition guarantees that this replacement carries a lagrangian structure equivalent to the original inclusion. The resulting symplectic quotient is a quasi-smooth 0-shifted symplectic derived Lie groupoid and is literally the homotopy pullback of the two lagrangians. Its classical loci recover the ordinary reduced spaces, and when the moment map is transverse to","pith_inferences":["If weak symplectic linearizability turns out to hold for every orbit of a proper quasi-symplectic groupoid, the theorem would supply a general smooth derived model for all singular symplectic reductions; the paper verifies the condition in compact and proper cases but leaves the general case open.","The derived quotient's tangent complex resembles the classical homological reduction complex, so a testable extension is whether that classical complex's symplectic form is the infinitesimal shadow of the reduced derived form under a differentiation comparison.","The lagrangian-correspondence viewpoint suggests that the symplectic strata of the reduced space can be organized as lagrangian correspondences into the derived quotient, a stratified refinement the paper gestures toward in its epilogue.","A direct test of the framework is to compute the derived quotient in an explicit quasi-hamiltonian or Poisson-Lie example where the moment map is critical, and verify that the 0-shifted form is nondegenerate on the derived locus."],"forward_implications":["Singular symplectic reduction at a critical level is not abandoned: under linearizability, the reduced object is a genuine 0-shifted symplectic derived Lie groupoid, with a nondegenerate reduced form even at singular points.","The reduced derived groupoid is a homotopy pullback of two lagrangian morphisms, so non-transverse intersections are replaced by well-defined derived intersections.","Shifted lagrangian correspondences compose in the smooth setting when levelwise transverse, giving a concrete realization of the symplectic category in which composition is geometrically controlled.","The framework unifies reduction at critical values across hamiltonian actions, group-valued moment maps, Poisson-Lie moment maps, and proper symplectic groupoids.","Classical reduced spaces appear as the classical locus of the derived quotient; in the transverse case the derived quotient is equivalent to the classical quotient by a levelwise weak equivalence."],"fun_headline_variants":["Derived groupoids repair singular symplectic reduction","Symplectic reduction via derived geometry, even when non-transverse","Homotopy pullback yields symplectic quotient for singular moment maps","Linearizable orbits give nondegenerate reduced symplectic forms","Derived Lie groupoids unify reduction at critical values"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole derived reduction theorem rests on the existence, at the chosen orbit, of a weak symplectic linearization: a local model satisfying three compatibility equations; the paper establishes this for proper groupoids, quasi-hamiltonian units, and coboundary Poisson-Lie cases, but not in general, and states that the hypothesis could not yet be eliminated.","fun_headline_variants_meta":{"raw":{"variants":["Derived groupoids repair singular symplectic reduction","Symplectic reduction via derived geometry, even when non-transverse","Homotopy pullback yields symplectic quotient for singular moment maps","Linearizable orbits give nondegenerate reduced symplectic forms","Derived Lie groupoids unify reduction at critical values"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000152,"raw_usage":{"total_tokens":961,"prompt_tokens":588,"completion_tokens":373,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":332,"completion_tokens_details":{"reasoning_tokens":287}},"tokens_in":332,"tokens_out":373,"duration_ms":4123,"temperature":1.0,"reasoning_tokens":287,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T18:11:27.292484+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a 1-shifted symplectic Lie groupoid and an orbit whose normal-bundle groupoid admits no weak symplectic linearization: concretely, check whether the pullback of the 2-form can be written as a linear part plus a simplicial coboundary with the prescribed restriction to the orbit. A single orbit where this fails shows the theorem does not apply; alternatively, compute the tangent-complex pairing at a classical point of the derived quotient in a critical example and check whether the map to the shifted cotangent complex is a quasi-isomorphism.","supporting_citations":[],"review_version":1}