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arxiv: 2606.02758 · v1 · pith:JTVI4RTSnew · submitted 2026-06-01 · 🧮 math.DG · cs.LG· math.CT

Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks

Michael Astwood This is my paper

Pith reviewed 2026-06-28 12:22 UTC · model grok-4.3

classification 🧮 math.DG cs.LGmath.CT
keywords Lie groupoidLie algebroidequivariant CNNcategory theorynatural transformationsconvolution layersglobal pooling
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The pith

Lie groupoid equivariant neural networks are built from lifting convolutions and convolution layers that are equivalent to Lie algebroid versions for suitable groupoids.

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

The paper specializes topological category-equivariant neural networks to the differentiable setting using Lie groupoids. It constructs these networks from Lie groupoid lifting convolutions and convolution layers. For suitable Lie groupoids, these networks are equivalent to certain Lie algebroid-equivariant neural networks. It also generalizes group invariant global pooling to groupoids and shows that the layers are special cases of admissible category-equivariant layers through continuous natural transformations between feature functors.

Core claim

Lie groupoid equivariant neural networks consist of Lie groupoid lifting convolutions and Lie groupoid convolution layers. For suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. Each layer defines continuous natural transformations between continuous feature functors, making them special cases of admissible category-equivariant layers. Groupoid invariant global pooling is introduced as a generalization of group invariant global pooling.

What carries the argument

Lie groupoid lifting convolutions and Lie groupoid convolution layers that induce continuous natural transformations between continuous feature functors.

If this is right

  • Networks equivariant to Lie groupoid actions can be built layer by layer using the defined convolutions.
  • Equivalence allows transferring results between groupoid and algebroid equivariant networks under the suitable condition.
  • Groupoid invariant global pooling applies to a wider class of symmetries than standard group pooling.
  • All such layers fit into the broader framework of admissible category-equivariant layers.

Where Pith is reading between the lines

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

  • The equivalence might enable using algebroid differential geometry tools to analyze or optimize groupoid-based networks.
  • This approach could be tested on specific Lie groupoids arising from manifolds with symmetries to verify practical equivariance.
  • Extending the construction beyond suitable groupoids may require additional continuity checks.

Load-bearing premise

Specializing topological category-equivariant networks to differentiable Lie groupoids preserves the necessary continuity and naturality without further restrictions.

What would settle it

A specific Lie groupoid and layer construction where the induced map between feature functors is not continuous or natural, violating the equivalence or special case property.

Figures

Figures reproduced from arXiv: 2606.02758 by Michael Astwood.

Figure 1
Figure 1. Figure 1: Diagram of a Lie groupoid G•, showing a point g ∈ G1 along with its source and target, as well as source and target fibers s −1 (x), t−1 (x) for several values of x ∈ G0. Dashed lines correspond to source fibers, while dotted lines correspond to target fibers. The solid central line corresponds to G0 embedded within G1 via the unit map u. Example 1 (Groupoid Defined by a Group). A group G can be used to de… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of a Lie groupoid equivariant neural network architecture, not including [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: a) Diagram of a manifold X = G0, showing the fibers Ex of a vector bundle defining feature fibers of constant dimension 2 over associated spaces Ω(x) = {x}. b) Diagram of a stratified space X with stratification X = (X \ Σ) ⊔ Σ, where the feature fibers are again defined over associated spaces Ω(x) = {x}, and above y ∈ Σ the fiber is 1-dimensional while the fiber above x ∈ X \ Σ is 2-dimensional. Definitio… view at source ↗
read the original abstract

We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.

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

0 major / 3 minor

Summary. The manuscript introduces Lie groupoid equivariant neural networks as a specialization of topological category-equivariant neural networks to the differentiable setting. Networks are built from Lie groupoid lifting convolutions and Lie groupoid convolution layers; for suitable Lie groupoids these are shown equivalent to certain Lie algebroid-equivariant networks. Groupoid invariant global pooling is presented as a generalization of group invariant global pooling. Each layer is shown to define continuous natural transformations between continuous feature functors and is therefore a special case of admissible category-equivariant layers.

Significance. If the equivalences and naturality statements hold, the work supplies a rigorous categorical and differential-geometric foundation that unifies several strands of equivariant network design. The explicit reduction to admissible category-equivariant layers and the generalization of global pooling are concrete contributions that could facilitate transfer of results across geometric deep learning frameworks.

minor comments (3)
  1. [abstract and §3] The qualifier 'suitable Lie groupoids' appears repeatedly (abstract, §1, §3) without an explicit characterization or list of sufficient conditions; a dedicated paragraph or lemma stating the precise hypotheses under which the equivalence holds would improve readability and verifiability.
  2. [§2 and §4] Notation for the lifting convolution and the feature functors is introduced in §2 but the continuity requirements on the functors are only stated informally; adding a short remark on the topology used on the space of sections would clarify the natural-transformation claim in §4.
  3. [§1] The manuscript cites the topological category-equivariant framework but does not include a self-contained one-paragraph recap of the relevant definitions; a brief reminder would help readers who are not already familiar with that prior work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its potential contributions to unifying equivariant network frameworks, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper frames its central claims as explicit definitions and constructive demonstrations: Lie groupoid equivariant networks are introduced by specialization from topological category-equivariant networks, composed from lifting convolutions and convolution layers, shown equivalent to Lie algebroid versions for suitable groupoids, with groupoid invariant pooling as a generalization, and each layer verified as a special case of admissible category-equivariant layers by direct demonstration that they yield continuous natural transformations between continuous feature functors. No fitted parameters, statistical predictions, or self-referential reductions appear; the specialization is asserted to preserve continuity and naturality under the stated qualifier, rendering the chain self-contained without reliance on unverified external self-citations for load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the constructions appear to rest on background results from category theory and differential geometry whose status cannot be audited here.

pith-pipeline@v0.9.1-grok · 5629 in / 1251 out tokens · 26108 ms · 2026-06-28T12:22:56.859776+00:00 · methodology

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Reference graph

Works this paper leans on

42 extracted references · 20 canonical work pages · 1 internal anchor

  1. [1]

    Aintablian.Differentiation of higher groupoid objects in tangent categories

    L. Aintablian.Differentiation of higher groupoid objects in tangent categories. PhD thesis, Universit¨ atsund Landesbibliothek Bonn, 2025. URL https://bonndoc.ulb. uni-bonn.de/xmlui/handle/20.500.11811/13688

    2025

  2. [2]

    Aintablian and C

    L. Aintablian and C. Blohmann. Differentiable groupoid objects and their abstract 24 Lie algebroids.Applied Categorical Structures, 33(5):33, 2025. doi:10.1007/s10485-025- 09830-2

    work page doi:10.1007/s10485-025- 2025

  3. [3]

    Arsie and E

    A. Arsie and E. Frazzoli.Groupoids in Control Systems and the Reachability Problem for a Class of Quantized Control Systems with Nonabelian Symmetries, page 18–31. Springer Berlin Heidelberg, 2007. ISBN 9783540714934. doi:10.1007/978-3-540-71493- 4 5

    work page doi:10.1007/978-3-540-71493- 2007

  4. [4]

    Bodnar, F

    C. Bodnar, F. Frasca, Y. G. Wang, G. Mont´ ufar, P. Li` o, and M. M. Bronstein. Weisfeiler and Lehman go topological: Message passing simplicial networks. InProceedings of the 38th International Conference on Machine Learning (ICML), volume 139 ofProceedings of Machine Learning Research, 2021. URL https://proceedings.mlr.press/v139/ bodnar21a.html

    2021

  5. [5]

    M. M. Bronstein, J. Bruna, T. Cohen, and P. Veliˇ ckovi´ c.Geometric Deep Learn- ing: Grids, Groups, Graphs, Geodesics, and Gauges. ArXiv Preprint, 2021. doi:10.48550/arxiv.2104.13478

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2104.13478 2021

  6. [6]

    R. Brown. Th´ eorie de jauge et groupo¨ ıdes.Fundamenta Mathematicae, 171(1):1–30,

  7. [7]

    doi:10.4064/fm171-1-1

    work page doi:10.4064/fm171-1-1

  8. [8]

    A. J. Bruce. From L ∞-algebroids to higher Schouten/Poisson structures.Reports on Mathematical Physics, 67(2):157–177, 4 2011. ISSN 0034-4877. doi:10.1016/s0034- 4877(11)00010-3

    work page doi:10.1016/s0034- 2011

  9. [9]

    Bujel, Y

    K. Bujel, Y. Gideoni, C. K. Joshi, and P. Li` o. Group invariant global pooling.CoRR, abs/2305.19207, 2023. doi:10.48550/arXiv.2305.19207

    work page doi:10.48550/arxiv.2305.19207 2023

  10. [10]

    Cohen and M

    T. Cohen and M. Welling. Group equivariant convolutional networks. In M. F. Balcan and K. Q. Weinberger, editors,Proceedings of The 33rd International Conference on Machine Learning, volume 48 ofProceedings of Machine Learning Research, pages 2990–2999, New York, New York, USA, 6 2016. PMLR. URL https://proceedings. mlr.press/v48/cohenc16.html

    2016

  11. [11]

    Cohen, M

    T. Cohen, M. Weiler, B. Kicanaoglu, and M. Welling. Gauge equivariant convolu- tional networks and the icosahedral CNN. InInternational conference on Machine 25 learning, pages 1321–1330. PMLR, 2019. URL https://proceedings.mlr.press/ v97/cohen19d/cohen19d.pdf

    2019

  12. [12]

    Colombo and D

    L. Colombo and D. M. d. Diego. Second-order variational problems on Lie groupoids and optimal control applications.Discrete and Continuous Dynamical Systems, 36(11): 6023–6064, Aug. 2016. ISSN 1078-0947. doi:10.3934/dcds.2016064

    work page doi:10.3934/dcds.2016064 2016

  13. [13]

    de Haan.Machine learning with generalized symmetries

    P. de Haan.Machine learning with generalized symmetries. PhD thesis, Universiteit van Amsterdam, 2025. URL https://hdl.handle.net/11245.1/ f4bff619-cb1e-46e7-8f5e-aeb7c4d40dd9

    2025

  14. [14]

    Dehmamy, R

    N. Dehmamy, R. Walters, Y. Liu, D. Wang, and R. Yu. Automatic symmetry dis- covery with lie algebra convolutional network. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, editors,Advances in Neural Infor- mation Processing Systems, volume 34, pages 2503–2515. Curran Associates, Inc.,

  15. [15]

    URL https://proceedings.neurips.cc/paper_files/paper/2021/file/ 148148d62be67e0916a833931bd32b26-Paper.pdf

    2021

  16. [16]

    Einizade, D

    A. Einizade, D. Thanou, F. D. Malliaros, and J. H. Giraldo. Continuous simplicial neural networks. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2026. URLhttps://openreview.net/forum?id=fPPfFMVTZo

    2026

  17. [17]

    uller, David I. and Schuh, Daniel , title =

    M. Favoni, A. Ipp, D. I. M¨ uller, and D. Schuh. Lattice gauge equivariant convolu- tional neural networks.Physical Review Letters, 128(3), Jan. 2022. ISSN 1079-7114. doi:10.1103/physrevlett.128.032003

    work page doi:10.1103/physrevlett.128.032003 2022

  18. [18]

    Finzi, S

    M. Finzi, S. Stanton, P. Izmailov, and A. G. Wilson. Generalizing convolutional neural networks for equivariance to Lie groups on arbitrary continuous data. InProceedings of the 37th International Conference on Machine Learning, ICML’20. JMLR, 2020. URLhttps://proceedings.mlr.press/v119/finzi20a/finzi20a.pdf

    2020

  19. [19]

    D. H. Fremlin.Measure theory: Set-theoretic measure theory v. 5-I. Torres Fremlin, Colchester, England, dec 2008

    2008

  20. [20]

    Grabowska, J

    K. Grabowska, J. Grabowski, M. Ku´ s, and G. Marmo. Lie groupoids in information geometry.Journal of Physics A: Mathematical and Theoretical, 52(50):505202, Nov

  21. [21]

    doi:10.1088/1751-8121/ab542e

    ISSN 1751-8121. doi:10.1088/1751-8121/ab542e. 26

    work page doi:10.1088/1751-8121/ab542e

  22. [22]

    D. Harel. On folk theorems.Communications of the ACM, 23(7):379–389, 7 1980. ISSN 1557-7317. doi:10.1145/358886.358892

    work page doi:10.1145/358886.358892 1980

  23. [23]

    Kock.Synthetic Geometry of Manifolds

    A. Kock.Synthetic Geometry of Manifolds. Cambridge University Press, Nov. 2009. ISBN 9780511691690. doi:10.1017/cbo9780511691690

    work page doi:10.1017/cbo9780511691690 2009

  24. [24]

    Kubarski

    J. Kubarski. Exponential mapping for Lie groupoids. InColloquium Mathematicum, volume 47, pages 267–282. Institute of Mathematics Polish Academy of Sciences, 1982. URLhttps://eudml.org/doc/264774

    1982

  25. [25]

    Mac Lane.Categories for the Working Mathematician, volume 5 ofGraduate Texts in Mathe- matics

    S. Mac Lane.Categories for the Working Mathematician. Springer New York, 1978. ISBN 9781475747218. doi:10.1007/978-1-4757-4721-8

    work page doi:10.1007/978-1-4757-4721-8 1978

  26. [26]

    K. C. H. Mackenzie.General Theory of Lie Groupoids and Lie Algebroids. Cambridge University Press, June 2005. ISBN 9780521499286. doi:10.1017/cbo9781107325883

    work page doi:10.1017/cbo9781107325883 2005

  27. [27]

    Maruyama

    Y. Maruyama. Categorical equivariant deep learning: Category-equivariant neu- ral networks and universal approximation theorems.ArXiv Preprint, 2025. doi:10.48550/arXiv.2511.18417

    work page doi:10.48550/arxiv.2511.18417 2025

  28. [28]

    Maruyama

    Y. Maruyama. Categorical trace loop networks for gauge-randomized holonomy regres- sion. InICLR 2026 Workshop on Geometry-grounded Representation Learning and Generative Modeling, 2026. URLhttps://openreview.net/forum?id=pH1KuDsEIL

    2026

  29. [29]

    Opperman, E

    B. Opperman, E. Alonso, and E. Mondragon. Groupoid-based internal state represen- tations for reinforcement learning with local symmetries. InProceedings of the 25th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2026), Workshop on Adaptive and Learning Agents (ALA), Paphos, Cyprus, May 2026. URLhttps://openaccess.city.ac.uk/i...

    2026

  30. [30]

    Rayat and G.-W

    A. Rayat and G.-W. Chern. Graph neural networks in the Wilson loop representation of abelian lattice gauge theories, 2026. URLhttps://arxiv.org/abs/2605.03901

    Pith/arXiv arXiv 2026

  31. [31]

    Rayat, Y

    A. Rayat, Y. Li, and G.-W. Chern. Gauge-equivariant graph neural networks for lattice gauge theories, 2026. URLhttps://arxiv.org/abs/2604.20797. 27

    Pith/arXiv arXiv 2026

  32. [32]

    Rom´ an and J

    A. Rom´ an and J. Villatoro. Convolution algebras of double groupoids and strict 2-groups.Symmetry, Integrability and Geometry: Methods and Applications, 10 2024. ISSN 1815-0659. doi:10.3842/sigma.2024.093

    work page doi:10.3842/sigma.2024.093 2024

  33. [33]

    V. G. Satorras, E. Hoogeboom, and M. Welling. E(n) equivariant graph neural networks. In M. Meila and T. Zhang, editors,Proceedings of the 38th International Conference on Machine Learning, volume 139 ofProceedings of Machine Learning Research, pages 9323–9332. PMLR, 18–24 Jul 2021. URL https://proceedings.mlr.press/v139/ satorras21a.html

    2021

  34. [34]

    V. G. Satorras, E. Hoogeboom, and M. Welling. E(n) equivariant graph neural networks. InProceedings of the 38th International Conference on Machine Learning (ICML), volume 139 ofProceedings of Machine Learning Research, pages 9323–9332, 2021. URL https://proceedings.mlr.press/v139/satorras21a.html

    2021

  35. [35]

    The Graph Neural Network Model

    F. Scarselli, M. Gori, A. C. Tsoi, M. Hagenbuchner, and G. Monfardini. The graph neural network model.IEEE Transactions on Neural Networks, 20(1):61–80, 2009. doi:10.1109/TNN.2008.2005605

    work page doi:10.1109/tnn.2008.2005605 2009

  36. [36]

    Schreiber, L

    U. Schreiber, L. Alfonsi, J. Lopez, and Anonymous. Lie infinity-algebroid, April

  37. [37]

    Non Peer- Reviewed Article

    URL https://ncatlab.org/nlab/show/Lie+infinity-algebroid. Non Peer- Reviewed Article. Accessed: May 31, 2026

    2026

  38. [38]

    Tang et al

    C. Tang et al. Deepscnn: A simplicial convolutional neural network for deep learning. Applied Intelligence, 55(4):281, 2025. doi:10.1007/s10489-024-06121-6

    work page doi:10.1007/s10489-024-06121-6 2025

  39. [39]

    O. M. Velarde, L. C. Parra, P. Boldi, and H. A. Makse. The role of fibration symmetries in geometric deep learning.Proceedings of the National Academy of Sciences, 123(4): e2416552123, 2026. doi:10.1073/pnas.2416552123

    work page doi:10.1073/pnas.2416552123 2026

  40. [40]

    D. P. Williams. Haar systems on equivalent groupoids.Proceedings of the American Mathematical Society, Series B, 3(1):1–8, 2016. ISSN 2330-1511. doi:10.1090/bproc/22. URLhttp://www.ams.org/bproc/2016-03-01/S2330-1511-2016-00022-6/

    work page doi:10.1090/bproc/22 2016

  41. [41]

    C. Zhu. n-groupoids and stacky groupoids.International Mathematics Research Notices,

  42. [42]

    doi:10.1093/imrn/rnp080

    ISSN 1687-0247. doi:10.1093/imrn/rnp080. 28

    work page doi:10.1093/imrn/rnp080