# Groupoid

A **groupoid** is a small category in which every morphism is an isomorphism; equivalently, it is a set equipped with an inversion operation \( g \mapsto g^{-1} \) and a partial binary operation \( (g,h) \mapsto gh \) satisfying axioms that generalize the group axioms by allowing multiplication to be defined only for some pairs of elements.<sup>[1](https://encyclopediaofmath.org/wiki/Groupoid)</sup> The term was introduced by Heinrich Brandt in 1926, and the concept has since become a standard tool in algebra, differential geometry, and topology.<sup>[1](https://encyclopediaofmath.org/wiki/Groupoid)</sup>

| Key fact | Detail |
|---|---|
| Definition | A small category in which every morphism is an isomorphism; equivalently a set with inversion and a partial binary operation<sup>[1](https://encyclopediaofmath.org/wiki/Groupoid)</sup> |
| Origin | Introduced by H. Brandt, "Über eine Verallgemeinerung des Gruppenbegriffes", Mathematische Annalen 96 (1926), pp. 360–366<sup>[1](https://encyclopediaofmath.org/wiki/Groupoid)</sup> |
| Smooth version | A Lie groupoid is a groupoid internal to smooth manifolds, with source and target maps surjective submersions<sup>[2](https://ncatlab.org/nlab/show/Lie%2Bgroupoid)</sup> |
| Introduced for smooth case | Charles Ehresmann, late 1950s, motivated by differential equations<sup>[3](https://publish.illinois.edu/ruiloja/math-595-lie-groupoids-and-lie-algebroids-fall-2021/)</sup> |
| Relation to groups | A group is a groupoid with a single object; Lie groupoids generalize smooth Lie group actions, with Lie groups at one end of the spectrum and plain manifolds at the other<sup>[4](https://pure.uva.nl/ws/files/28325413/Thesis.pdf)</sup> |
| Orbit spaces | The objects of a Lie groupoid decompose into leaves (orbits), and orbit spaces describe singular spaces such as orbifolds and leaf spaces of foliations<sup>[5](https://academic.oup.com/imrn/article/2021/2/1249/5289644)</sup> |
| Operator algebras | Jean Renault's construction assigns a C*-algebra to a locally compact groupoid; amenability implies the full and reduced C*-algebras coincide<sup>[6](https://library.impa.br/cbm/27CBM/27CBM_05.pdf)</sup> |

## Algebraic definition

A groupoid may be presented in two equivalent ways. Categorically, it is a (small) category in which every morphism is an isomorphism. Algebraically, it is a set \( G \) with a unary operation \( g \mapsto g^{-1} \) and a partial binary operation \( (g,h) \mapsto gh \) such that:<sup>[1](https://encyclopediaofmath.org/wiki/Groupoid)</sup>

1. \( gg^{-1} \) and \( g^{-1}g \) are always defined;
2. \( gh \) is defined if and only if \( g^{-1}g = hh^{-1} \);
3. if \( gh \) and \( hk \) are defined, then \( (gh)k \) and \( g(hk) \) are defined and equal;
4. each of \( g^{-1}gh \), \( hg^{-1}g \), \( gg^{-1}h \), and \( hgg^{-1} \) equals \( h \) whenever it is defined.

The partiality of the multiplication is the essential difference from a group. In the categorical picture, elements are arrows from a source object to a target object, and two arrows can be composed if and only if the second starts where the first ends.<sup>[3](https://publish.illinois.edu/ruiloja/math-595-lie-groupoids-and-lie-algebroids-fall-2021/)</sup>

## History

Brandt introduced the notion in a paper submitted on 12 December 1925 and published in *Mathematische Annalen*, volume 96, pages 360–366, under the title "Über eine Verallgemeinerung des Gruppenbegriffes" ("On a generalization of the group concept").<sup>[7](https://mathshistory.st-andrews.ac.uk/Extras/Preston_semigroups/)</sup> His version, now called a Brandt groupoid, dealt with finite structures and satisfied postulates in which a product \( ab = c \) determines each of the three elements from the other two, together with one-sided identities and inverses for each element.<sup>[7](https://mathshistory.st-andrews.ac.uk/Extras/Preston_semigroups/)</sup><sup> • </sup><sup>[8](https://www.sciencedirect.com/topics/mathematics/occupied-cell)</sup>  Brandt's structure, viewed in modern terms, is what has since been called either a Brandt groupoid or a Brandt semigroup, and one history of the subject credits Brandt with starting the development of groupoid theory in 1927.<sup>[7](https://mathshistory.st-andrews.ac.uk/Extras/Preston_semigroups/)</sup><sup> • </sup><sup>[9](https://www.ams.org/bookstore/pspdf/surv-241-intro.pdf)</sup>

## Lie groupoids

A **Lie groupoid** is a groupoid internal to the category of smooth manifolds: the space of arrows \( X_1 \) and the space of objects \( X_0 \) are both smooth manifolds, all structure maps (source \( s \), target \( t \), unit \( u \), inversion \( i \), and multiplication \( m \)) are smooth, and the source and target maps are surjective submersions.<sup>[2](https://ncatlab.org/nlab/show/Lie%2Bgroupoid)</sup><sup> • </sup><sup>[10](https://people.se.cmich.edu/watts1j/math6230s17/groupoids.pdf)</sup>  The submersion condition on \( s \) and \( t \) is not decorative: it guarantees that the space \( X_1 \times_{s,t} X_1 \) of composable pairs of arrows is itself a smooth manifold, so that the smoothness requirement on multiplication makes sense.<sup>[2](https://ncatlab.org/nlab/show/Lie%2Bgroupoid)</sup><sup> • </sup><sup>[11](https://francisrlb.com/2026/09/02/lie-groupoids-and-differential-equations/)</sup>  Some definitions allow the arrow space to be non-Hausdorff.<sup>[12](https://www.math.tecnico.ulisboa.pt/~rfern/Estudantes/TeseLuisAlex.pdf)</sup>

Lie groupoids were introduced by [Charles Ehresmann](https://www.edgepedia.org/charles-ehresmann) in the late 1950s as a way of describing smooth symmetries of a smooth family of objects, where Lie groups describe smooth symmetries of a single object; his original motivation came from differential equations.<sup>[3](https://publish.illinois.edu/ruiloja/math-595-lie-groupoids-and-lie-algebroids-fall-2021/)</sup> The concept interpolates between two familiar structures: a [Lie group](https://www.edgepedia.org/lie-group) acting on a one-point space is a Lie groupoid, and so is an ordinary smooth manifold with no arrows beyond the units.<sup>[4](https://pure.uva.nl/ws/files/28325413/Thesis.pdf)</sup>

**Orbits and properness.** The objects of a Lie groupoid \( \mathcal{G} \rightrightarrows M \) split into equivalence classes called leaves or orbits, where two objects are equivalent when some arrow has one as source and the other as target; the quotient is the orbit space or leaf space. Many singular spaces in geometry arise this way, including orbifolds, leaf spaces of foliations, and moduli spaces of curves.<sup>[5](https://academic.oup.com/imrn/article/2021/2/1249/5289644)</sup> A Lie groupoid is **proper** when the map \( (s,t) : \mathcal{G} \to M \times M \) is a proper map (and the groupoid is Hausdorff); proper Lie groupoids are the groupoid analog of proper Lie group actions and share many of their good properties, such as linearization along an orbit.<sup>[5](https://academic.oup.com/imrn/article/2021/2/1249/5289644)</sup><sup> • </sup><sup>[13](https://link.springer.com/article/10.1007/s11005-017-1011-6)</sup>  For a Lie group \( G \) regarded as a one-object groupoid, properness is equivalent to compactness of \( G \); an action groupoid is proper exactly when the underlying Lie group action is proper.<sup>[13](https://link.springer.com/article/10.1007/s11005-017-1011-6)</sup>

## Applications

**Foliation theory.** [André Haefliger](https://www.edgepedia.org/andre-haefliger) introduced transversal structures to foliations in the 1970s using the holonomy groupoid of a foliated manifold, an approach that has been central to the development of the subject.<sup>[3](https://publish.illinois.edu/ruiloja/math-595-lie-groupoids-and-lie-algebroids-fall-2021/)</sup><sup> • </sup><sup>[14](https://paperzz.com/doc/8802628/ecm-2000)</sup>

**Noncommutative geometry and index theory.** Lie groupoids entered noncommutative geometry through [Alain Connes](https://www.edgepedia.org/alain-connes)'s work in the 1980s; a key element of his strategy for proving the [Atiyah–Singer index theorem](https://www.edgepedia.org/atiyah-singer-index-theorem) was the tangent groupoid he attached to a space, and this line led to the Connes–Skandalis index theory for foliations.<sup>[3](https://publish.illinois.edu/ruiloja/math-595-lie-groupoids-and-lie-algebroids-fall-2021/)</sup>

**Poisson geometry.** Motivated by quantization, Vladimir Karasev and Alan Weinstein introduced the symplectic groupoid of a Poisson manifold in the late 1980s, as a way to untangle the symplectic foliation underlying the Poisson structure.<sup>[3](https://publish.illinois.edu/ruiloja/math-595-lie-groupoids-and-lie-algebroids-fall-2021/)</sup>

**Algebraic geometry.** [Alexander Grothendieck](https://www.edgepedia.org/alexander-grothendieck) introduced stacks in the late 1960s via fibered categories over a site, and fibered categories can be viewed as a type of sheaf of groupoids; this line of development led to the concept of a gerbe.<sup>[3](https://publish.illinois.edu/ruiloja/math-595-lie-groupoids-and-lie-algebroids-fall-2021/)</sup>

**Stacks and orbispaces.** Differentiable stacks are geometric objects that have manifolds and orbifolds as special instances, and they can be presented as the transverse geometry of a Lie groupoid; two Lie groupoids presenting the same object are considered equivalent.<sup>[15](https://arxiv.org/abs/1212.6714)</sup><sup> • </sup><sup>[16](https://ems.press/journals/pm/articles/12156)</sup>

## Groupoid C*-algebras

To a locally compact groupoid equipped with a Haar system, Jean Renault's construction associates a [C*-algebra](https://www.edgepedia.org/c-algebra), generalizing the group C*-algebra.<sup>[17](https://www.utgjiu.ro/math/sma/v01/a08.html)</sup> For a locally compact étale groupoid \( G \), the full C*-algebra \( C^*(G) \) is the completion of the convolution algebra \( C_c(G) \) with respect to the full norm, and the reduced C*-algebra \( C^*_{\mathrm{red}}(G) \) is its quotient defined through the regular representation; the matrix algebra \( M_n(\mathbb{C}) \) arises this way from the pair groupoid on \( n \) points.<sup>[6](https://library.impa.br/cbm/27CBM/27CBM_05.pdf)</sup> A property called amenability, generalizing amenability of groups, implies the equality \( C^*(G) = C^*_{\mathrm{red}}(G) \); examples where the two coincide include the Cuntz groupoids, tail equivalence relations, and the irrational rotation groupoid.<sup>[6](https://library.impa.br/cbm/27CBM/27CBM_05.pdf)</sup> Amenable groupoids produce nuclear C*-algebras, mirroring the discrete-group situation.<sup>[18](https://web.math.ku.dk/~rordam/students/Munkholm-Fagprojekt.pdf)</sup>

## Structure theory

A Lie groupoid is **regular** when its leaves all have the same dimension. Moerdźik's structure theorem shows that a regular Lie groupoid fits into an exact sequence \( K \to \mathcal{G} \to E \), where \( K \) is a bundle of Lie groups and \( E \) is a foliation groupoid; when \( \mathcal{G} \) is proper, \( E \) is an orbifold groupoid and the fibers of \( K \) are compact Lie groups.<sup>[5](https://academic.oup.com/imrn/article/2021/2/1249/5289644)</sup> In general a proper Lie groupoid may have leaves of varying dimension, and its orbit space can be more singular than an orbifold; nevertheless the orbit space of a proper Lie groupoid is a Gromov–Hausdorff limit of orbifolds, and such orbit spaces can be triangulated.<sup>[5](https://academic.oup.com/imrn/article/2021/2/1249/5289644)</sup><sup> • </sup><sup>[19](https://arxiv.org/abs/1101.0180)</sup>

## References

1. [Groupoid, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Groupoid)
2. [Lie groupoid, nLab](https://ncatlab.org/nlab/show/Lie%2Bgroupoid)
3. [Math 595 – Lie Groupoids and Lie Algebroids, Rui Loja Fernandes, University of Illinois](https://publish.illinois.edu/ruiloja/math-595-lie-groupoids-and-lie-algebroids-fall-2021/)
4. [PhD thesis chapter on Lie groupoids, Universiteit van Amsterdam](https://pure.uva.nl/ws/files/28325413/Thesis.pdf)
5. [Resolutions of Proper Riemannian Lie Groupoids, International Mathematics Research Notices](https://academic.oup.com/imrn/article/2021/2/1249/5289644)
6. [C*-algebras and Dynamical Systems, IMPA 27th Brazilian Mathematics Colloquium](https://library.impa.br/cbm/27CBM/27CBM_05.pdf)
7. [Gordon Preston on semigroups, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Extras/Preston_semigroups/)
8. [Occupied Cell overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/occupied-cell)
9. [Introduction to groupoids and their roles, AMS Surveys 241 introduction](https://www.ams.org/bookstore/pspdf/surv-241-intro.pdf)
10. [Lie Groupoids lecture notes, Central Michigan University](https://people.se.cmich.edu/watts1j/math6230s17/groupoids.pdf)
11. [Lie groupoids and differential equations, Francis Bischoff](https://francisrlb.com/2026/09/02/lie-groupoids-and-differential-equations/)
12. [MSc thesis definition of Lie groupoid, Instituto Superior Técnico, Lisbon](https://www.math.tecnico.ulisboa.pt/~rfern/Estudantes/TeseLuisAlex.pdf)
13. [Orbispaces as differentiable stratified spaces, Letters in Mathematical Physics](https://link.springer.com/article/10.1007/s11005-017-1011-6)
14. [ECM 2000 notes on holonomy groupoids and convolution algebras](https://paperzz.com/doc/8802628/ecm-2000)
15. [Lie Groupoids and Differentiable Stacks, arXiv:1212.6714](https://arxiv.org/abs/1212.6714)
16. [Matias L. del Hoyo, Lie groupoids and their orbispaces, Portugaliae Mathematica 70 (2013)](https://ems.press/journals/pm/articles/12156)
17. [Mădălina Roxana Buneci, Groupoid C*-algebras, Surveys in Mathematics and its Applications 1 (2006), 71–98](https://www.utgjiu.ro/math/sma/v01/a08.html)
18. [Groupoid C*-algebras project notes, University of Copenhagen](https://web.math.ku.dk/~rordam/students/Munkholm-Fagprojekt.pdf)
19. [Geometry of orbit spaces of proper Lie groupoids, arXiv:1101.0180](https://arxiv.org/abs/1101.0180)

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