Pith. sign in

REVIEW 2 major objections 5 minor 48 references

Fundamentals of Lie categories and Yang-Mills theory for multiplicative Ehresmann connections

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Yang-Mills theory can be built for multiplicative Ehresmann connections on any Lie algebroid, dropping transitivity and integrability.

desk verdict Genuinely new Yang-Mills framework for non-transitive Lie algebroids, but Theorem 13's variation computation is not displayed and needs referee scrutiny before the central claim becomes citable. read the letter →

arxiv 2507.08220 v2 pith:2EYWILLA submitted 2025-07-10 math.DG

Žan Grad This is my paper · ORCID
classification math.DG MSC 22A2253C0553C0870S15
keywords LiecategoriesgroupoidsalgebroidsmultiplicativeEhresmannconnectionsYang-MillstheoryprimitiveIMbundlesofidealsS^1-bundlegerbes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that Yang-Mills theory, usually formulated on principal bundles over a fixed spacetime, can be built for much more general geometric objects: arbitrary Lie groupoids and Lie algebroids, with no transitivity or integrability assumptions. The replacement for a principal connection is a multiplicative Ehresmann connection, or infinitesimally an IM connection, and the paper develops the calculus needed to vary such connections. Its central result is a pair of gauge-invariant field equations, one longitudinal and one transversal, whose solutions are exactly the critical points of a natural action on the class of primitive connections; in the classical transitive case the pair collapses to the usual Yang-Mills equation. A separate, shorter part of the paper drops invertibility in Lie groupoids to found the theory of Lie categories, motivated by irreversible processes in thermodynamics.

What carries the argument

The load-bearing object is the primitive IM connection. An IM connection is an infinitesimal multiplicative 1-form $(C,v)$ on a Lie algebroid $A$ with values in a bundle of ideals $k$, whose second component $v: A \to k$ splits the short exact sequence $0 \to k \to A \to A/k \to 0$; it is primitive when its curvature $\Omega_{(C,v)}$ is cohomologically trivial, $\Omega_{(C,v)}=\delta_0 F$, so the curvature is encoded by a base 2-form $F$ called the curving. Around this object the paper builds the horizontal exterior covariant derivative $D_{(C,v)}=h^* d_\nabla$ on the Weil complex, the affine deformation formula for curvature, the curvature 3-form $G=d_\nabla F$, the ad-invariant metric, and the adaptedness condition; together these convert the variational problem into the two equations of Theorem 13 and force the curving to be a Laplacian eigenvector.

What would settle it

Take a concrete non-transitive Lie algebroid with a nontrivial centre, for instance an action algebroid with non-constant isotropy, and compute whether any primitive adapted IM connection exists there; if none exist, or if the only solutions of the two field equations have vanishing curvature, then the proposed relaxation of transitivity is vacuous for that class of algebroids.

Watch

Extended reading notes

Core claim

The central claim is a positive answer to the question posed in the introduction: can Yang-Mills theory be constructed for multiplicative Ehresmann connections? The answer is yes, and Theorem 13 states it precisely. Fix a bundle of ideals $k$ in a Lie algebroid $A$ and a primitive IM connection $(C,v)$ with curving $F$, meaning the curvature equals $\delta_0 F$ for a base 2-form $F$. With an ad-invariant metric and the adaptedness condition, the connection is longitudinally critical for the action $S((C,v),F)=\int_M \langle F,F\rangle_k + \mu\int_M \langle G,G\rangle_k$, with $G=d_\nabla F$, exactly when $d_\nabla \star F=0$, and transversally critical exactly when $d_\nabla \star G=\tfrac{1}{\mu}\star F$. Together with the Bianchi identities $d_\nabla F=G$ and $d_\nabla G=0$, these equations describe gauge field dynamics along and transverse to the orbit foliation; for transitive algebroids $G$ vanishes and the first equation alone recovers classical Yang-Mills theory. The framework specialises to a Yang-Mills theory for $S^1$-bundle gerbes.

Load-bearing premise

The load-bearing restriction is that the variational theory is defined only on primitive IM connections, connections whose curvature is completely captured by a base 2-form called the curving, that also satisfy the adaptedness condition, so if such connections are rare for a given algebroid, the generalized Yang-Mills theory has a small or empty field space.

Editorial extensions

If this is right

  • In the transitive case the 3-curvature vanishes and the pair of equations collapses to the classical Yang-Mills equation $d_\nabla \star F=0$, so principal-bundle Yang-Mills is recovered as a special case.
  • The theory is gauge invariant: the action is unchanged by pullback along inner automorphisms of an integrating groupoid and by infinitesimal inner derivations, so the equations have the symmetry expected of a gauge theory.
  • Any solution has its curving $F$ as an eigenvector of the covariant Laplacian, $\Delta F = -\tfrac{1}{\mu}F$, replacing the classical harmonicity condition $\Delta F = 0$.
  • Non-integrable and non-transitive Lie algebroids become legitimate backgrounds for variational gauge theory, so gauge fields can live on singular orbit foliations rather than only on manifolds with a transitive symmetry algebroid.
  • The example of central $S^1$-extensions yields a Yang-Mills theory for $S^1$-bundle gerbes, connecting the framework to higher geometry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • This goes beyond the paper: if primitivity and adaptedness are rare, the framework's field space is small; computing this locus for concrete non-transitive algebroids would tell how many new systems the theory actually covers.
  • This goes beyond the paper: the eigenvalue equation $\Delta F = -\tfrac{1}{\mu}F$ invites a spectral reading, so one could ask whether solution moduli are deformation invariants of the algebroid.
  • This goes beyond the paper: since adaptedness is flatness in the transitive case, the genuinely new regime is high codimension with nontrivial centre; explicit adapted primitive connections there would be the sharpest test of physical content.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This PhD thesis develops two projects. The first part introduces Lie categories, i.e. internal categories in smooth manifolds, and studies their cores, two Lie algebroids, ranks, extensions to groupoids, completeness of invariant vector fields, and an application to statistical thermodynamics. The second and principal part develops the infinitesimal and global theory of multiplicative Ehresmann connections on bundles of ideals, constructs horizontal exterior covariant derivatives on the Bott–Shulman–Stasheff and Weil complexes, proves van Est commutativity at the level of multiplicative forms, and then proposes a Yang–Mills theory on the space of primitive IM connections. The central claim, stated in the Introduction and in Theorem 13, is that the multiplicative Yang–Mills action S((C,v),F)=∫⟨F,F⟩+µ∫⟨G,G⟩, with G=d∇F, has critical points characterized by d∇⋆F=0 and, under an adaptedness condition, by d∇⋆G=(1/µ)⋆F, yielding a pair of gauge-invariant equations that relax the transitivity assumption of classical Yang–Mills theory.

Significance. The paper contains several substantial independent contributions: the deformation formulae for curvatures of multiplicative and IM connections (Theorems 11–12), the explicit construction of the horizontal exterior covariant derivative on Weil cochains (Theorems 7–8), the obstruction classes for invariant connections and multiplicative connections, and a systematic treatment of Lie categories. If Theorem 13 is correct, the proposed framework is a genuine generalization of Yang–Mills theory to non-transitive, non-integrable algebroids and gives a variational theory for S^1-bundle gerbes. However, the central variational claim is precisely where the manuscript is weakest: the longitudinal variation of the µ∫⟨G,G⟩ term appears to contribute a nonvanishing term that is not accounted for in the stated Euler–Lagrange equivalence. The paper is also honest about the restrictive nature of primitivity and adaptedness, but this restriction has not been quantified outside the examples listed. I therefore regard the manuscript as valuable but not yet established in its main claim.

major comments (2)
  1. [§5.3.2(iii), §5.3.3, Theorem 13] The proof of the first equivalence in Theorem 13 is not supported by the displayed deformation formulae. The longitudinal variation is taken along (C,v)+λδ0γ with curving Fλ = F + λd∇γ − λ²/2[γ,γ], and the action contains µ∫⟨G,G⟩ with G=d∇F. Even before any change of the induced connection is computed, the first-order derivative of the second term is 2µ∫_M ⟨G, d∇d∇γ⟩, which after integration by parts equals ±2µ∫_M ⟨δ∇G, d∇γ⟩. This term is not identically zero in the intended regime: ∇ need not be flat and G is allowed to be nonzero. The manuscript displays no cancellation identity that would make this term vanish or absorb it into the variation of ∇. As written, the equivalence “longitudinal criticality ⇔ d∇⋆F=0” therefore requires an additional argument, namely either a proof that the µ-term does not contribute on the deformation directions or a corrected Euler–Lagrange equation containing the 3-curvature contribution.
  2. [§5.3.2, §5.3.4, Introduction] The advertised scope “relaxing the transitivity condition” is conditioned on a potentially very thin domain: the action is defined only on primitive IM connections, i.e. those with Ω(C,v)=δ0F, and transversal criticality requires the additional adaptedness condition. The author states that adaptedness is “rather strong” and that in the transitive case it is equivalent to flatness. Nonemptiness is shown for transitive algebroids, semisimple typical fibres, and central S^1-extensions, but no genericity statement is given for arbitrary algebroids. If primitive, adapted connections are rare outside the listed examples, the central claim covers fewer new systems than the introductory formulation suggests. The manuscript should either prove an existence statement for a non-transitive, non-integrable family with G≠0 or explicitly delimit the range of the theory.
minor comments (5)
  1. [§1.4] Proposition 1.29 states that the left and right Lie algebroids of a Lie category are Lie algebroids, but the proof is omitted with the remark that it is the same as in the groupoid case; please include a reference or a short argument, since this is the foundation of the rank and completeness results.
  2. [§2.3, equations (C.1)–(C.3)] The compatibility conditions (C.1)–(C.3) are labelled with the prefix “C.” but they are part of Example 2.21 and are used later without a displayed equation number; renumber them in the main sequence.
  3. [Introduction, symmetries table] The table lists δ∇G = −(1/µ)F while Theorem 13 states d∇⋆G=(1/µ)⋆F; the sign and Hodge-star convention should be unified and explained once, preferably immediately after the action is defined.
  4. [§5.2.1] The foliated theory is said to require “some extra assumptions” and “regularity assumptions” on the orbit foliation, but these assumptions are not stated explicitly in the Introduction or in the statement of Theorem 10; please spell them out for the reader.
  5. [§2.3, Remark 2.19] The term “curved double complex” is used in several places before it is defined in the Introduction; please define it at first use or give a forward pointer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Yang–Mills equations are derived from the action by first-variation computations, and the primitive/adapted restrictions are openly imposed domain restrictions, not hidden fits or self-citation chains.

full rationale

The central claim is Theorem 13, which derives the paired equations d∇⋆F=0 and d∇⋆G=(1/µ)⋆F as critical-point equations for S((C,v),F)=∫⟨F,F⟩+µ∫⟨G,G⟩, with G=d∇F, on the explicitly defined class of primitive IM connections with curving F. The action is not fitted to the equations: G is defined as d∇F, and the equations are obtained by varying the curving and connection through the deformation formulae of Theorems 11–12. The restriction to primitive IM connections, defined in §5.3.2 by Ω(C,v)=δ0F, and the adaptedness condition are presented openly as defining the domain and as a 'rather strong' additional constraint; they do not by themselves assert the equations. The theorem is conditional: within the primitive, adapted class, criticality is equivalent to the displayed field equations. This is a constrained variational problem, not a case in which the conclusion is an input. The paper's self-citations (to [38], [39], [48], [51]) are to the author's own prior work and to standard references, but the load-bearing infinitesimal operator D(C,v) and the variation formulae are established inside the thesis, so no central claim reduces to an unverified self-citation. The skeptic concern that the µ∫⟨G,G⟩ term contributes a d∇d∇γ term to the longitudinal variation is a potential correctness gap in the proof of the first equivalence, but it is not a circularity: it does not exhibit an equation that equals its own input by construction. Similarly, the admitted scarcity of primitive/adapted connections, and the statement in the Introduction that in the transitive case adaptedness is equivalent to flatness, narrow the applicability of the theorem but do not make the derivation circular. No circular step satisfying the required quote-and-reduction standard was found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 3 invented entities

The ledger shows the theory's reach is bought with explicit structural choices. The only numerical free parameter is µ, but the framework also requires non-numerical choices: an ad-invariant metric on the isotropy bundle, a Riemannian metric and orientation on the base, the primitive-connection restriction, and adaptedness. The axioms are mostly standard machinery from the cited literature (van Est theory [16], VB-calculus [36, 37], transversality with corners [58]), with two ad hoc restrictions specific to this paper (primitive connections and adaptedness). The invented objects (curving F, curvature 3-form G, the horizontal derivative D(C,v)) are derived constructs with internal consistency checks rather than entities pulled from a hat. Overall the contribution is genuine, but the boundary between what is derived and what is chosen is wider than the abstract suggests.

free parameters (4)
  • mu (structure constant) = arbitrary real constant
    Appears in the multiplicative Yang-Mills action S = ∫⟨F,F⟩ + µ∫⟨G,G⟩ and in the second equation d∇⋆G = (1/µ)⋆F. Chosen by hand; the paper notes µ can be chosen arbitrarily.
  • ad-invariant metric on isotropy bundle k
    The action functional requires ⟨·,·⟩k on k = ker ρ satisfying ad-invariance; existence is assumed, not proven, and fails for typical fibres whose isotropy Lie algebras are not reductive with invariant bilinear forms.
  • Riemannian metric and orientation on M
    Needed to define the Hodge star ⋆F and ⋆G in the equations; chosen by hand.
  • foliated-theory regularity assumptions
    Theorem 10 assumes A is a regular Lie algebroid 'with some extra assumptions' and an orientable M; the extra assumptions are postponed to §5.2.
assumptions (5)
  • standard math Standard theorems of Lie groupoid/algebroid theory, the van Est theorem for representation-valued forms (Theorem 2.22 from [16]), and VB-groupoid/VB-algebroid calculus [36, 37].
    Invoked throughout Chapters 2-4 as background from [16, 25, 36, 37, 48, 51].
  • standard math The Bott-Shulman-Stasheff and Weil complexes with representation coefficients and their simplicial differentials are well-defined and satisfy δ² = 0.
    Recalled in §2.3 from [16]; the entire cochain-map arguments in Chapters 3-4 rest on these complexes.
  • ad hoc to paper The domain of the multiplicative Yang-Mills action is restricted to primitive IM connections, i.e., those with cohomologically trivial curvature Ω(C,v) = δ0F.
    Introduced in §5.3.2 as 'a crucial insight'; this restriction is not inherited from classical Yang-Mills and is required for the variational formulation to close.
  • ad hoc to paper Adaptedness is imposed for transversal criticality (in the µ = -1 case, δ0(δ∇G) = Ω(C,v)).
    Stated in Theorem 13 and the surrounding discussion; the author concedes this is 'rather strong' and forces flatness in the transitive case.
  • domain assumption Existence of multiplicative Ehresmann connections for the bundle of ideals, obstructed by classes in H^{2,1}(G;k)_Hor and H^{2,1}(A;k)_Hor.
    Chapter 4 constructs obstruction classes; the whole framework presupposes the vanishing case, i.e., that such connections exist for the algebroids considered.
invented entities (3)
  • Curving F and curvature 3-form G of a primitive IM connection independent evidence
    purpose: The curving F is a base 2-form with δ0F = Ω(C,v); G = d∇F is the curvature 3-form that appears in the second Yang-Mills equation and carries the transversal dynamics.
    G is not pulled from a hat: it satisfies the Bianchi-derived identities d∇G = 0 and δ0G = 0, and in the transitive case G = 0 recovers classical Yang-Mills; the S^1-bundle gerbe example (§5.3.6) provides an independent context where these objects reproduce known structures.
  • Multiplicative Yang-Mills equations as a coupled pair independent evidence
    purpose: d∇⋆F = 0 and d∇⋆G = (1/µ)⋆F describe longitudinal and transversal stationarity of the action.
    Derived from the variational principle with stated assumptions; reduces to d∇⋆F = 0 when G = 0 and yields self-dual/anti-self-dual solution classes in 5 dimensions (§5.3.5), giving checkable consequences.
  • Horizontal subcomplexes and horizontal exterior covariant derivative D(C,v) on Weil cochains independent evidence
    purpose: Needed to define curvature of IM connections and the infinitesimal Bianchi identity; Definition 4.31 and Theorem 4.53.
    The operator is pinned down by the cochain-map property δD = Dδ (Theorem 8) and by van Est compatibility at the level of multiplicative forms (Theorem 9), which are internal consistency checks.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fundamentals of Lie categories and Yang-Mills theory for multiplicative Ehresmann connections." pith.science (2026). https://pith.science/paper/2EYWILLA

@misc{pith2026250708220,
  author       = {Pith},
  title        = {Pith review of: Fundamentals of Lie categories and Yang-Mills theory for multiplicative Ehresmann connections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2EYWILLA}},
  note         = {Machine review of arXiv:2507.08220}
}
abstract

The first and shorter part of this thesis deals with the structural assumption of invertibility in a Lie groupoid. When this assumption is dropped, we obtain the notion of a Lie category: a small category, endowed with a compatible differentiable structure. We introduce various examples of Lie categories, examine their differences and similarities with Lie groupoids, and research the notions emerging naturally from the lack of invertibility of arrows. The aim of the second and principal part of this thesis is to provide a far-reaching generalization of Yang-Mills theory, extending it from the classical setting of principal bundles to general Lie groupoids and algebroids. The notion of a principal bundle connection is now replaced with that of a more general multiplicative Ehresmann connection. In obtaining this generalization, we make various advances to the theory of such connections, as well as invariant linear connections on representations. We develop the obstruction classes for their existence, generalize the (horizontal) exterior covariant derivative to the representation-valued Bott-Shulman-Stasheff and Weil complexes, and inspect their relationship with the van Est map. We research the class of multiplicative connections with cohomologically trivial curvature, which are central to obtaining the desired generalization. Applying the variational principle to this framework rests upon our developed formulae for affine deformations of multiplicative connections. Ultimately, we develop the extension of Yang-Mills theory to a non-integrable and non-transitive setting: the classical Yang-Mills equation is upgraded to a gauge-invariant pair of equations, which now describe the dynamics of gauge fields in both the longitudinal and transversal directions with respect to the (singular) orbit foliation. As an example, we obtain a Yang-Mills theory for $S^1$-bundle gerbes.

Figures

Figures reproduced from arXiv: 2507.08220 by the authors.

Figure 1.5
Figure 1.5. The order category of R. Although simple, this example has an important property: C can be seen as the preimage of the set [0, ∞) under the functor f : R × R → (R, +), f(y, x) = y − x. The following example is a generalization of this; we will make use of it when considering appli￾cations to statistical thermodynamics in §1.8. Example 1.11. Suppose a Lie category D ⇒ X without boundary is given, together with a smoo… view at source ↗
Figure 1.8
Figure 1.8. The disjoint union of the order category with the pair groupoid. [PITH_FULL_IMAGE:figures/full_fig_p031_1_8.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 31 canonical work pages

  1. [22]

    Marius Crainic, Rui Loja Fernandes, and David Martínez Torres,Regular Poisson manifolds of compact types, Astérisque 413 (2019), viii + 154, DOI 10.24033/ast (English, with English and French summaries)

    doi:10.24033/ast 2019

  2. [23]

    217, American Mathematical Society, Providence, RI, [2021]©2021

    Marius Crainic, Rui Loja Fernandes, and Ioan Mărcuţ,Lectures on Poisson geometry, Graduate Studies in Mathematics, vol. 217, American Mathematical Society, Providence, RI, [2021]©2021

    work page 2021

  3. [24]

    Noncommut

    Marius Crainic and João Nuno Mestre,Measures on differentiable stacks, J. Noncommut. Geom.13 (2019), no. 4, 1235–1294, DOI 10.4171/jncg/362

    doi:10.4171/jncg/362 2019

  4. [25]

    Marius Crainic, Maria Amelia Salazar, and Ivan Struchiner,Multiplicative forms and Spencer operators, Math. Z. 279 (2015), no. 3-4, 939–979, DOI 10.1007/s00209-014-1398-z

    work page doi:10.1007/s00209-014-1398-z 2015

  5. [26]

    Reine Angew

    Matias del Hoyo and Rui Loja Fernandes,Riemannian metrics on Lie groupoids, J. Reine Angew. Math.735 (2018), 143–173, DOI 10.1515/crelle-2015-0018. MR3757473

    doi:10.1515/crelle-2015-0018 2018

  6. [27]

    Korean Math

    Andrzej Derdzinski,Harmonic curvature in dimension four, J. Korean Math. Soc.62 (2025), no. 1, 217–252, DOI 10.4134/JKMS.j240001. MR4847263

    doi:10.4134/jkms.j240001 2025

  7. [28]

    1156, Springer, Berlin, 1985, pp

    , Riemannian manifolds with harmonic curvature, Global differential geometry and global analysis 1984 (Berlin, 1984), Lecture Notes in Math., vol. 1156, Springer, Berlin, 1985, pp. 74–85, DOI 10.1007/BFb0075087. MR0824063

    work page doi:10.1007/bfb0075087 1984

  8. [29]

    Thiago Drummond and Leandro Egea,Differential forms with values in VB-groupoids and its Morita invari- ance, J. Geom. Phys.135 (2019), 42–69, DOI 10.1016/j.geomphys.2018.08.019

    work page doi:10.1016/j.geomphys.2018.08.019 2019

Show all 48 references
  1. [30]

    Drummond, M

    T. Drummond, M. Jotz Lean, and C. Ortiz,VB-algebroid morphisms and representations up to homotopy, Differential Geom. Appl.40 (2015), 332–357, DOI 10.1016/j.difgeo.2015.03.005. MR3333112

    2015 doi

  2. [31]

    J. J. Duistermaat and J. A. C. Kolk,Lie groups, Universitext, Springer-Verlag, Berlin, 2000

    2000

  3. [32]

    Charles Ehresmann, Catégories topologiques et catégories différentiables, Colloque Géom. Diff. Globale (Bruxelles, 1958), Librairie Universitaire, Louvain, 1959, pp. 137–150 (French)

    1958

  4. [33]

    Preprint available at arXiv:2210.02924

    Simon-Raphael Fischer, Integrating curved Yang-Mills gauge theories (2023). Preprint available at arXiv:2210.02924

    2023 arXiv

  5. [34]

    An intro- duction

    Theodore Frankel,The geometry of physics, 3rd ed., Cambridge University Press, Cambridge, 2012. An intro- duction. MR2884939

    2012

  6. [35]

    Alfonso Garmendia and Sylvie Paycha,Principal bundle groupoids, their gauge group and their nerve, J. Geom. Phys. 191 (2023), Paper No. 104865, 21, DOI 10.1016/j.geomphys.2023.104865. MR4598921

    2023

  7. [36]

    Math.223 (2010), no

    Alfonso Gracia-Saz and Rajan Amit Mehta,Lie algebroid structures on double vector bundles and representa- tion theory of Lie algebroids, Adv. Math.223 (2010), no. 4, 1236–1275, DOI 10.1016/j.aim.2009.09.010

    2010 doi

  8. [37]

    Symplectic Geom

    , VB-groupoids and representation theory of Lie groupoids, J. Symplectic Geom. 15 (2017), no. 3, 741–783, DOI 10.4310/JSG.2017.v15.n3.a5

    2017 doi

  9. [38]

    Preprint available at arXiv:2503.08873

    Žan Grad,Covariant derivatives in the representation-valued Bott-Shulman-Stasheff and Weil complex(2025). Preprint available at arXiv:2503.08873

    2025 arXiv

  10. [39]

    Noncommut

    , Fundamentals of Lie categories , J. Noncommut. Geom. 19 (2025), no. 1, 211–248, DOI 10.4171/jncg/563

    2025 doi

  11. [40]

    Victor Guillemin, Eugene Lerman, and Shlomo Sternberg,Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge, 1996

    1996

  12. [41]

    Victor Guillemin and Alan Pollack,Differential topology, AMS Chelsea Publishing, Providence, RI, 2010

    2010

  13. [42]

    Mark J. D. Hamilton,Mathematical gauge theory, Universitext, Springer, Cham, 2017

    2017

  14. [43]

    28, Oxford University Press, Oxford, 2019

    Chris Heunen and Jamie Vicary, Categories for quantum theory, Oxford Graduate Texts in Mathematics, vol. 28, Oxford University Press, Oxford, 2019

    2019

  15. [44]

    MR3726907

    Jürgen Jost, Riemannian geometry and geometric analysis, 7th ed., Universitext, Springer, Cham, 2017. MR3726907

    2017

  16. [45]

    Jotz,Obstructions to representations up to homotopy and ideals, Asian J

    M. Jotz,Obstructions to representations up to homotopy and ideals, Asian J. Math.26 (2022), no. 2, 137–166, DOI 10.4310/ajm.2022.v26.n2.a1

    2022 doi

  17. [46]

    Kosmann-Schwarzbach and K

    Y. Kosmann-Schwarzbach and K. C. H. Mackenzie,Differential operators and actions of Lie algebroids, Quan- tization, Poisson brackets and beyond (Manchester, 2001), Contemp. Math., vol. 315, Amer. Math. Soc., Providence, RI, 2002, pp. 213–233, DOI 10.1090/conm/315/05482

    2001 doi

  18. [47]

    Ruis Kuwabara, On spectra of the Laplacian on vector bundles, J. Math. Tokushima Univ.16 (1982), 1–23. MR0691445

    1982

  19. [48]

    Math.220 (2009), no

    Camille Laurent-Gengoux, Mathieu Stiénon, and Ping Xu,Non-abelian differentiable gerbes, Adv. Math.220 (2009), no. 5, 1357–1427, DOI 10.1016/j.aim.2008.10.018. 152

    2009 doi

  20. [49]

    Lee,Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol

    John M. Lee,Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013

    2013

  21. [50]

    David Li-Bland and Eckhard Meinrenken,On the van Est homomorphism for Lie groupoids, Enseign. Math. 61 (2015), no. 1-2, 93–137, DOI 10.4171/LEM/61-1/2-5

    2015 doi

  22. [51]

    Math.427 (2023), Paper No

    Rui Loja Fernandes and Ioan Mărcuţ,Multiplicative Ehresmann connections, Adv. Math.427 (2023), Paper No. 109124, 84, DOI 10.1016/j.aim.2023.109124

    2023

  23. [52]

    Preprint available at arXiv:2205.11457

    , Poisson geometry around Poisson submanifolds(2023). Preprint available at arXiv:2205.11457

    2023 arXiv

  24. [53]

    Kirill C. H. Mackenzie,Duality and triple structures, The breadth of symplectic and Poisson geometry, Progr. Math., vol. 232, Birkhäuser Boston, Boston, MA, 2005, pp. 455–481, DOI 10.1007/0-8176-4419-9_15

    2005 doi

  25. [54]

    Reine Angew

    , Ehresmann doubles and Drinfel’d doubles for Lie algebroids and Lie bialgebroids, J. Reine Angew. Math. 658 (2011), 193–245, DOI 10.1515/CRELLE.2011.092

    2011 doi

  26. [55]

    213, Cambridge University Press, Cambridge, 2005

    , General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005

    2005

  27. [56]

    Kirill C. H. Mackenzie and Ping Xu,Lie bialgebroids and Poisson groupoids, Duke Math. J.73 (1994), no. 2, 415–452, DOI 10.1215/S0012-7094-94-07318-3

    1994 doi

  28. [57]

    213 (2014), no

    Ioan Mărcuţ, Rigidity around Poisson submanifolds , Acta Math. 213 (2014), no. 1, 137–198, DOI 10.1007/s11511-014-0118-1. MR3261013

    2014 doi

  29. [58]

    173, North-Holland Publishing Co., Amsterdam, 1992

    Juan Margalef Roig and Enrique Outerelo Domínguez,Differential topology, North-Holland Mathematics Stud- ies, vol. 173, North-Holland Publishing Co., Amsterdam, 1992

    1992

  30. [59]

    Varghese Mathai and David Roberts, Yang-Mills theory for bundle gerbes, J. Phys. A 39 (2006), no. 20, 6039–6044, DOI 10.1088/0305-4470/39/20/027. MR2238132

    2006 doi

  31. [60]

    Dusa McDuff and Dietmar Salamon,Introduction to symplectic topology, 2nd ed., Oxford Mathematical Mono- graphs, The Clarendon Press, Oxford University Press, New York, 1998

    1998

  32. [61]

    Math.67 (2021), no

    Eckhard Meinrenken,On the integration of transitive Lie algebroids, Enseign. Math.67 (2021), no. 3-4, 423– 454, DOI 10.4171/lem/1015

    2021 doi

  33. [62]

    Eckhard Meinrenken and Jeffrey Pike,The Weil algebra of a double Lie algebroid, Int. Math. Res. Not. IMRN 11 (2021), 8550–8622, DOI 10.1093/imrn/rnz361

    2021 doi

  34. [63]

    Michor,Topics in differential geometry, Graduate Studies in Mathematics, vol

    Peter W. Michor,Topics in differential geometry, Graduate Studies in Mathematics, vol. 93, American Math- ematical Society, Providence, RI, 2008

    2008

  35. [64]

    Notes available at arXiv:math/0212266

    Ieke Moerdijk,Introduction to the language of stacks and gerbes, 2002. Notes available at arXiv:math/0212266

    2002 arXiv

  36. [65]

    Ieke Moerdijk and Janez Mrčun,On integrability of infinitesimal actions, Amer. J. Math.124 (2002), no. 3, 567–593

    2002

  37. [67]

    M. K. Murray, Bundle gerbes , J. London Math. Soc. (2) 54 (1996), no. 2, 403–416, DOI 10.1093/acprof:oso/9780199534920.003.0012. MR1405064

    1996

  38. [68]

    Sussmann,Orbits of families of vector fields and integrability of distributions, Trans

    Héctor J. Sussmann,Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188, DOI 10.2307/1996660

    1973 doi

  39. [69]

    Warner,Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol

    Frank W. Warner,Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR0722297

    1983

  40. [70]

    Alan Weinstein,Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.)16 (1987), no. 1, 101–104, DOI 10.1090/S0273-0979-1987-15473-5. 153

    1987 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.