pith. sign in

arxiv: 2605.17351 · v1 · pith:5YQGCY3Bnew · submitted 2026-05-17 · 🧮 math.DG · math.SG

Automorphisms of Lie groupoids and symplectic reduction on orbifolds

Bohui Chen , Cheng-Yong Du , Fengyu Jiang This is my paper

Pith reviewed 2026-05-19 22:58 UTC · model grok-4.3

classification 🧮 math.DG math.SG
keywords Lie groupoidssymplectic reductionorbifoldsHamiltonian actionsKan fibrationsLie 2-groupoidsautomorphism 2-groups
0
0 comments X

The pith

Symplectic reductions of Hamiltonian étale Lie 2-group actions on orbifolds generally produce symplectic Lie 2-groupoids and remain orbifolds under an isotropic free condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first builds the automorphism 2-group BAut(X) for any Lie groupoid X and shows that homomorphisms from a 2-group G into BAut(X) are equivalent to Kan fibrations over G with fiber X. This equivalence supplies a precise way to define actions of étale Lie 2-groups on orbifolds and to formulate Hamiltonian actions of those 2-groups via the same Kan-fibration data. The authors then apply the construction to symplectic reduction: the reduced space is always a symplectic Lie 2-groupoid, and it remains an ordinary orbifold precisely when the action satisfies an isotropic free condition. The same machinery also yields a slice theorem for ordinary group actions on Lie groupoids.

Core claim

The reduction of a symplectic orbifold by a Hamiltonian action of an étale Lie 2-group, formulated through a Kan fibration over the 2-group with fiber the orbifold, is in general a symplectic Lie 2-groupoid; under an additional isotropic free condition the reduced object is still an orbifold. The same framework produces a slice theorem for ordinary group actions on Lie groupoids.

What carries the argument

The 2-group BAut(X) of automorphisms of a Lie groupoid X, whose homomorphisms from G are equivalent to Kan fibrations over G with fiber X and which therefore encodes 2-group actions and their Hamiltonian reductions.

If this is right

  • Symplectic reduction by étale Lie 2-group actions produces symplectic Lie 2-groupoids in the general case.
  • Under an isotropic free condition the same reduction yields an ordinary symplectic orbifold.
  • A slice theorem holds for ordinary group actions on Lie groupoids.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Kan-fibration description may let researchers import techniques from higher category theory directly into symplectic reduction on singular spaces.
  • One could test whether the resulting 2-groupoids admit natural quantizations that recover known orbifold quantum cohomology rings.
  • The slice theorem might extend to give local normal forms for 2-group actions, allowing explicit computations of reduced spaces near fixed points.

Load-bearing premise

That Hamiltonian actions of étale Lie 2-groups can be defined via Kan fibrations over the 2-group and that the symplectic form descends appropriately to the reduction quotient.

What would settle it

An explicit Hamiltonian action of an étale Lie 2-group on a symplectic orbifold whose reduced space fails to carry a compatible 2-groupoid structure or whose symplectic form fails to descend.

read the original abstract

In this paper, the 2-group BAut(X) of automorphisms of a Lie groupoid X is constructed. Considering the 2-group G action on X, we explain the equivalence between 2-group homomorphisms from G to BAut(X) with Kan fibrations over G with fiber X. This justifies the notion of Kan fibration for 2-group actions on Lie groupoids. As an application, we formulate Hamiltonian actions of \'etale Lie 2-groups on orbifolds in terms of Kan fibrations and study the symplectic reductions. We show that, in general, the reduction is in fact a symplectic Lie 2-groupoid, and under certain isotropic free condition, the reduction is still an orbifold. Also the slice theorem of a group G action on Lie groupoids is proved.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs the 2-group BAut(X) of automorphisms of a Lie groupoid X and establishes an equivalence between 2-group homomorphisms from an étale Lie 2-group G to BAut(X) and Kan fibrations over G with fiber X. It applies this framework to formulate Hamiltonian actions of étale Lie 2-groups on orbifolds via Kan fibrations and studies the associated symplectic reductions, claiming that the reduction is in general a symplectic Lie 2-groupoid and remains an orbifold under an isotropic free condition. The paper also proves a slice theorem for ordinary group actions on Lie groupoids.

Significance. If the descent of the symplectic structure is rigorously established, the work would provide a higher-categorical approach to symplectic reduction on orbifolds, extending standard Lie groupoid techniques. The equivalence between 2-group homomorphisms and Kan fibrations supplies a categorical foundation for defining actions, while the slice theorem adds a concrete result on group actions that could be useful in geometric applications.

major comments (2)
  1. [Application to symplectic reductions] Application section on symplectic reductions: the claim that the reduction is a symplectic Lie 2-groupoid requires that the original symplectic form is invariant under the étale Lie 2-group action (formulated via the Kan fibration) and descends to a closed non-degenerate 2-form on the quotient Lie 2-groupoid. The manuscript invokes this descent without an explicit invariance argument or non-degeneracy verification on the reduced object, which is load-bearing for the central reduction theorem.
  2. [Formulation of Hamiltonian actions] Formulation of Hamiltonian actions: the definition via Kan fibrations over the 2-group with fiber the orbifold groupoid must be shown to preserve the necessary moment-map and isotropy conditions so that the isotropic-free hypothesis indeed yields an orbifold; without this link the reduction statements do not automatically follow from the general case.
minor comments (2)
  1. [Abstract] Abstract: the typesetting of 'étale' contains a stray backslash; ensure consistent use of accented characters throughout.
  2. [Definitions and constructions] Notation section: the precise construction of BAut(X) as a 2-group and the fiberwise action on the Lie groupoid X should be stated with explicit source/target maps to avoid ambiguity when comparing to standard automorphism groupoids.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major point below, providing clarifications from the existing arguments in the paper and indicating the revisions we will make to strengthen the exposition in the application section.

read point-by-point responses

  1. Referee: Application section on symplectic reductions: the claim that the reduction is a symplectic Lie 2-groupoid requires that the original symplectic form is invariant under the étale Lie 2-group action (formulated via the Kan fibration) and descends to a closed non-degenerate 2-form on the quotient Lie 2-groupoid. The manuscript invokes this descent without an explicit invariance argument or non-degeneracy verification on the reduced object, which is load-bearing for the central reduction theorem.

    Authors: We agree that making the invariance and descent arguments fully explicit will improve readability. The Hamiltonian action is defined via the Kan fibration, which by construction (using the equivalence with 2-group homomorphisms to BAut(X) established in Section 3) ensures that the étale Lie 2-group action preserves the symplectic form on the underlying orbifold groupoid. The closedness of the descended form follows from the fact that the original form is closed and the quotient map is a submersion in the appropriate sense for Lie 2-groupoids. Non-degeneracy on the reduced object holds in general by the properties of the symplectic reduction procedure for Lie groupoids, with the isotropic-free condition additionally guaranteeing that the result remains an orbifold. In the revised manuscript we will insert a new proposition in the application section that states these facts explicitly, together with a short verification of the descent of the 2-form and its non-degeneracy. revision: yes

  2. Referee: Formulation of Hamiltonian actions: the definition via Kan fibrations over the 2-group with fiber the orbifold groupoid must be shown to preserve the necessary moment-map and isotropy conditions so that the isotropic-free hypothesis indeed yields an orbifold; without this link the reduction statements do not automatically follow from the general case.

    Authors: The Kan fibration formulation is introduced precisely because the earlier equivalence theorem (between 2-group homomorphisms and Kan fibrations with fiber X) transfers the moment-map and isotropy data from the classical Hamiltonian action on the orbifold groupoid into the 2-categorical setting. Consequently, the isotropic-free condition on the action directly implies that the reduced object is an orbifold, as the fibers of the quotient map remain discrete. To make this link transparent, we will add a short lemma in the revised version that recalls how the moment map and isotropy conditions are encoded in the Kan fibration data and confirms that they are preserved under the reduction construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations build from standard constructions

full rationale

The paper constructs BAut(X) as the 2-group of automorphisms of a Lie groupoid X and establishes an equivalence between 2-group homomorphisms G to BAut(X) and Kan fibrations over G with fiber X. This equivalence is used to formulate Hamiltonian actions of étale Lie 2-groups on orbifolds. The symplectic reduction is then shown to yield a symplectic Lie 2-groupoid in general (and an orbifold under isotropic-free conditions), with a separate slice theorem for ordinary group actions. These steps rely on standard Lie groupoid theory and symplectic geometry without reducing any central claim to a fitted parameter, self-definition, or unverified self-citation chain. The abstract and context indicate independent content in the reduction arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard definitions of Lie groupoids, 2-groups, and symplectic structures on orbifolds; new entities like BAut(X) and Kan fibrations for actions are introduced without independent external evidence beyond the constructions themselves.

axioms (2)
  • standard math Lie groupoids are smooth categories internal to the category of manifolds with invertible arrows forming a manifold
    Invoked implicitly as the base object X whose automorphisms are studied
  • domain assumption Symplectic forms on orbifolds admit reduction under suitable groupoid actions preserving the structure
    Required for the reduction to remain symplectic
invented entities (1)
  • BAut(X) as a 2-group no independent evidence
    purpose: To encode the automorphisms of the Lie groupoid X in a higher categorical structure
    Constructed in the paper as the central new object

pith-pipeline@v0.9.0 · 5666 in / 1479 out tokens · 33384 ms · 2026-05-19T22:58:34.671676+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

  1. [1]

    A. Adem, J. Leida, and Y. Ruan.Orbifolds and Stringy Topology, volume 171 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2007. 22

    work page 2007

  2. [2]

    Behrend and E

    K. Behrend and E. Getzler. Geometric higher groupoids and categories. In J.-B. Bost, H. Hofer, F. Labourie, Y. L. Jan, X. Ma, and W. Zhang, editors,Geometry, Analysis and Probability, volume 310 ofProgress in Mathematics, pages 1–45. Springer International Publishing, 2017. 10

    work page 2017

  3. [3]

    Chen, C.-Y

    B. Chen, C.-Y. Du, and R. Wang. The groupoid structure of groupoid morphisms.Journal of Geometry and Physics, 145:103486, 2019. 2, 14

    work page 2019

  4. [4]

    Crainic and I

    M. Crainic and I. Struchiner. On the linearization theorem for proper Lie groupoids.Annales scien- tifiques de l’ ´Ecole normale sup´ erieure, 46(5):723–746, 2013. 36

    work page 2013

  5. [5]

    A. C. da Silva.Lectures on Symplectic Geometry. Number 1764 in Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2008. 43

    work page 2008

  6. [6]

    M. L. del Hoyo. Lie groupoids and their orbispaces.Portugaliae Mathematica, 70(2):161–209, 2013. 32

    work page 2013

  7. [7]

    Friedman

    G. Friedman. Survey article: an elementary illustrated introduction to simplicial sets.Rocky Mountain Journal of Mathematics, 42(2):353–423, 2012. 7

    work page 2012

  8. [8]

    S. I. Gelfand and Y. I. Manin.Methods of homological algebra. Springer Monographs in Mathematics. Springer, Berlin, 2003. 5

    work page 2003

  9. [9]

    P. G. Goerss and J. F. Jardine.Simplicial Homotopy Theory. Birkh¨ auser Basel, 2009. 5

    work page 2009

  10. [10]

    Henriques

    A. Henriques. Integratingl ∞-algebras.Compositio Mathematica, 144(4):1017–1045, 2008. 7

    work page 2008

  11. [11]

    Hoffman and R

    B. Hoffman and R. Sjamaar. Stacky Hamiltonian actions and symplectic reduction.International Math- ematics Research Notices, 2021(20):15209–15300, 2021. 4, 16, 40, 41, 42

    work page 2021

  12. [12]

    D. Li. Higher groupoid actions, bibundles, and differentiation.arXiv:1512.04209 [math.DG], 2015. 3, 7, 9, 16

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  13. [13]

    Marsden and A

    J. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry.Reports on Mathe- matical Physics, 5(1):121–130, 1974. 41

    work page 1974

  14. [14]

    Mendes and M

    R. Mendes and M. Radeschi. A slice theorem for singular Riemannian foliations, with applications. Transactions of the American Mathematical Society, 371(7):4931–4949, Nov. 2018. 36 AUTOMORPHISMS OF LIE GROUPOIDS AND SYMPLECTIC REDUCTION ON ORBIFOLDS 47

    work page 2018

  15. [15]

    K. R. Meyer. Symmetries and Integrals in Mechanics. In M. M. Peixoto, editor,Dynamical Systems, Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26–August 14, 1971, pages 259–272. Elsevier, 1973. 41

    work page 1971

  16. [16]

    Moerdijk and J

    I. Moerdijk and J. Mrˇ cun.Introduction to Foliations and Lie Groupoids, volume 91 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2003. 22

    work page 2003

  17. [17]

    M. J. Pflaum, H. Posthuma, and X. Tang. Geometry of orbit spaces of proper Lie groupoids.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2014(694):49–84, 2014. 36

    work page 2014

  18. [18]

    Trentinaglia and C

    G. Trentinaglia and C. Zhu. Strictification of ´ etale stacky Lie groups.Compositio Mathematica, 148(3):807–834, 2011. 5

    work page 2011

  19. [19]

    Weinstein

    A. Weinstein. Linearization problems for Lie algebroids and Lie groupoids.Letters in Mathematical Physics, 52(1):93–102, 2000. 36

    work page 2000

  20. [20]

    Zhu.n-groupoids and stacky Lie groupoids.International Mathematics Research Notices, 2009(21):4087–4141, 2009

    C. Zhu.n-groupoids and stacky Lie groupoids.International Mathematics Research Notices, 2009(21):4087–4141, 2009. 5, 9, 10

    work page 2009

  21. [21]

    N. T. Zung. Proper groupoids and momentum maps: Linearization, affinity, and convexity.Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure, 39(5):841–869, 2006. 36 Department of Mathematics and Yangtz center of Mathematics, Sichuan University, Chengdu 610065, China E-mail address: chenbohui@scu.edu.cn School of Mathematical Science, Sichuan Normal Uni...

    work page 2006