## From groups to groupoids: a brief survey (1987)

Venue: | Bull. London Math. Soc |

Citations: | 59 - 7 self |

### BibTeX

@ARTICLE{Brown87fromgroups,

author = {Ronald Brown},

title = {From groups to groupoids: a brief survey},

journal = {Bull. London Math. Soc},

year = {1987},

pages = {113--134}

}

### Years of Citing Articles

### OpenURL

### Abstract

A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range of applications, so that even for purely group-theoretical work, it can be useful to take a path through the world of groupoids.

### Citations

123 | On the algebra of cubes
- Brown, Higgins
- 1981
(Show Context)
Citation Context ... 2-dimensional Van Kampen type theorem for crossed modules [27] which yields some new homotopy computations in dimension 2 (see also [19]). Similar remarks apply to all dimensions, using ω -groupoids =-=[28, 29]-=-; these have interrelated structures in all dimensions, with n groupoid structures in dimension n, corresponding to the gluing of n-cubes in the n different directions. There is a fundamental ω -group... |

75 | Colimit theorems for relative homotopy groups
- BROWN, HIGGINS
- 1981
(Show Context)
Citation Context ... 2-dimensional Van Kampen type theorem for crossed modules [27] which yields some new homotopy computations in dimension 2 (see also [19]). Similar remarks apply to all dimensions, using ω -groupoids =-=[28, 29]-=-; these have interrelated structures in all dimensions, with n groupoid structures in dimension n, corresponding to the gluing of n-cubes in the n different directions. There is a fundamental ω -group... |

68 |
Versal deformations and algebraic
- Artin
- 1974
(Show Context)
Citation Context ... to be morphisms of C, where m is defined on the pull-back of s, t, which is assumed to exist. The axioms for a groupoid are expressed in a standard way using diagrams in C. For example, one finds in =-=[3, 64]-=- the definition of an algebraic groupoid as a groupoid object in the category of algebraic spaces. In fact the general notions of structured category and structured groupoid were defined and developed... |

56 | A survey of foliations and operator algebras
- Connes
- 1982
(Show Context)
Citation Context ... by a number of other writers (compare [124, 125]). There is for such groupoids a notion of convolution algebra,. and the resulting C∗–algebras have been powerfully exploited by A. Connes and ”others =-=[41]-=-. For example, they lead to an index theorem for foliations, generalising the Atiyah-Singer index theorem. The Introduction to [90] gives a succinct summary of the uses of groupoids in Connes’ theory.... |

50 | On the connection between the second relative homotopy groups of some related spaces
- Brown, Higgins
- 1978
(Show Context)
Citation Context ...categories with connection are applied to homotopy theory in [138]. One practical use of double groupoids is that they allow for a proof of a 2-dimensional Van Kampen type theorem for crossed modules =-=[27]-=- which yields some new homotopy computations in dimension 2 (see also [19]). Similar remarks apply to all dimensions, using ω -groupoids [28, 29]; these have interrelated structures in all dimensions,... |

46 |
Combinatorial group theory: a topological approach
- Cohen
- 1989
(Show Context)
Citation Context ...roof of a more powerful theorem; which can’t be all bad. The most general formulation to date of this theorem on the fundamental groupoid is in [36]. Other texts which have followed this approach are =-=[39, 78, 160]-=-. Somewhat earlier, Crowell and Fox in [43, p. 153] took the view that a few definitions like that of a group, or a topological space, have a fundamental importance for the whole of mathematics that c... |

34 |
Double groupoids and crossed modules, Cahiers Topologie G!eom. Diff!erent!ıelle Cat!egoriques 17
- Brown, Spencer
- 1976
(Show Context)
Citation Context ... how the theory of groups with an algebraic structure is a pale shadow of a rich theory of algebraically structured groupoids. Crossed modules are also equivalent to double groupoids with connections =-=[38]-=-. These model well the idea of using squares instead of paths, so that one can form compositions of the type (**) In this way double groupoids allow for ‘an algebraic inverse to subdivision’. It turns... |

29 |
Elements of Modern Topology
- Brown
- 1968
(Show Context)
Citation Context ...ced by the usual composition of paths: y x z 3 Applications of the fundamental groupoid My own introduction to the use of groupoids came with this last example, in 1965. I was writing a topology text =-=[14]-=-, which was to include the Van Kampen Theorem on the fundamental group of a union of spaces. I wanted a version of this theorem which would imply the determination of the fundamental group of the circ... |

27 | Groupoids and Van Kampen’s theorem - Brown - 1967 |

24 | Topology and Groupoids
- Brown
- 2006
(Show Context)
Citation Context ...s we have similar terminology to that for functors, namely faithful, full, representative, and also a variety of other types such as quotient, universal, covering [78], fibration, and discrete kernel =-=[15]-=-. See [45] for a discussion of congruences in groupoids. It may disturb people to learn that the first isomorphism theorem fails for groupoids. But in fact these apparent difficulties and complication... |

23 |
crossed modules and the fundamental groupoid of a topological group
- Brown, Spencer, et al.
- 1976
(Show Context)
Citation Context ...th zero multiplication. In general, group objects internal to the standard algebraic categories are ’abelian’ in some way. The situation is quite different for groupoid objects. A result published in =-=[37]-=-, but known much earlier, is that a groupoid object internal to groups, which we call here a cat 1 -group, is equivalent to a crossed module, which is a homomorphism µ : M → P of groups, together with... |

22 |
Kampen theorems for diagrams of spaces
- BROWN, LoDAY, et al.
- 1987
(Show Context)
Citation Context ...nce between the categories of cat n -groups and of ‘crossed n-cubes of groups’, thus giving a subtle, n-fold version of crossed modules. Loday and I have proved a van Kampen theorem for cat n -groups =-=[32, 33]-=-, which generalises the major part of the van Kampen theorem for ω -groupoids. The case n = 2 leads to some new algebraic constructions, such as a non-abelian tensor product M⊗N of groups M,N each of ... |

18 | Homotopical excision, and Hurewicz theorems, for n-cubes of spaces
- Brown, Loday
- 1987
(Show Context)
Citation Context ...werful approach to both fundamental groups and ideas of symmetry. Also, higher dimensional groupoids have led in homotopy theory to new results and calculations which seem unobtainable by other means =-=[19, 33, 34, 57]-=-. In view of the fundamental nature of our ideas of symmetry, I expect that multiple groupoids will lead to a formulation of ideas of ‘higher order symmetry’, or ‘symmetry of symmetries’ and methods o... |

17 |
Uber eine Verallgemeinerung des Gruppenbegries
- Brandt
- 1927
(Show Context)
Citation Context ... = ta, t(a −1 ) = sa, aa −1 = isa, a −1 a = ita. An element a is often written as an arrow a : sa → ta. � • a �• b �• • ab �• sa ta = sb tb sa tb Groupoids were introduced by Brandt in his 1926 paper =-=[11]-=-, although he always used the extra condition that for all x, y in Ob(G) there is an a in G such that sa = x, ta = y — such a groupoid we nowadays call connected or transitive. Brandt’s definition of ... |

15 | Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups’, Topology 23
- BROWN
- 1984
(Show Context)
Citation Context ...actical use of double groupoids is that they allow for a proof of a 2-dimensional Van Kampen type theorem for crossed modules [27] which yields some new homotopy computations in dimension 2 (see also =-=[19]-=-). Similar remarks apply to all dimensions, using ω -groupoids [28, 29]; these have interrelated structures in all dimensions, with n groupoid structures in dimension n, corresponding to the gluing of... |

13 |
Eléments de Mathématique. I. Théorie des Ensembles. Chapter III: Ensembles ordonnés, cardinaux, nombres entiers
- Bourbaki
- 1956
(Show Context)
Citation Context ...objects’-for a recent paper see [106]. A recent application of groupoids is in combinatorics by Joyal [86] using species (French: espèces, German: Gattungen) of structure. The term is due to Bourbaki =-=[5]-=-; its aim is to give a general description of the kind of structures which occur in mathematics, so there are species of structure of order, of topology, of vector space, of complex analytic manifold ... |

8 |
Higher dimensional group theory’, Low dimensional topology
- BROWN
- 1982
(Show Context)
Citation Context ...dules: The notion is due to J.H.C. Whitehead [149], the name being used in [150]. Surveys of their use and relationships with classical notions of homotopy theory and homological algebra are given in =-=[17,18,30,31]-=-. The equivalence between crossed modules and cat 1 -groups is given as follows. Let µ : M → P be a crossed module. Let G = P ⋊ M be the semi-direct product group, using the action of P on M, and let ... |

8 | The fundamental groupoid as a topological groupoid - Brown, Danesh-Naruie |

6 |
Excision homotopique en basse dimension
- BROWN, LODAY
- 1984
(Show Context)
Citation Context ...nce between the categories of cat n -groups and of ‘crossed n-cubes of groups’, thus giving a subtle, n-fold version of crossed modules. Loday and I have proved a van Kampen theorem for cat n -groups =-=[32, 33]-=-, which generalises the major part of the van Kampen theorem for ω -groupoids. The case n = 2 leads to some new algebraic constructions, such as a non-abelian tensor product M⊗N of groups M,N each of ... |

5 |
Simplicial T-complexes and crossed complexes
- Ashley, D
- 1978
(Show Context)
Citation Context ... G is a groupoid, then G is also a category, and so its nerve NG is defined. (In fact NG has more structure, namely it is a ‘simplicial T-complex of rank 1’, as shown by Dakin, [46]. See also Ashley, =-=[2]-=-.) The geometric realisation |NG| of the nerve of the groupoid G is called the classifying space BG of the groupoid G. It is a CW-complex, with one vertex for each element of Ob(G), one component for ... |

5 |
Groupoids and the Mayer-Vietoris sequence
- Brown, Heath, et al.
- 1983
(Show Context)
Citation Context ...roupoidπ1X [14]. Fibrations of groupoids [15] occur naturally in a number of ways in group or group action theory; the resulting exact sequences give results on the original group theoretic situation =-=[74, 25, 26]-=-. We should also refer to the neglected paper by P. A. Smith [137] where a covering morphism is called a regular homomorphism. One of the irritations of group theory is that the setHom(H,K) of homomor... |

5 |
A van Kampen theorem for unions of nonconnected spaces
- Brown, Salleh, et al.
- 1984
(Show Context)
Citation Context ...Theorem’ for U,V,W connected. So one obtains a simpler proof of a more powerful theorem; which can’t be all bad. The most general formulation to date of this theorem on the fundamental groupoid is in =-=[36]-=-. Other texts which have followed this approach are [39, 78, 160]. Somewhat earlier, Crowell and Fox in [43, p. 153] took the view that a few definitions like that of a group, or a topological space, ... |

4 |
Generator and relations for groups of homeomorphisms’, Transformation groups
- ABELS
- 1977
(Show Context)
Citation Context ...ver X (thus V is a fundamental domain for the action). The group Γ has generators [g] for g ∈ G such that V ∩ gV ̸= ∅, and relations [gh] = [g][h], whenever g, h ∈ G and V ∩ gV ∩ hgV ̸= ∅. The papers =-=[1,123]-=- show that this theorem is related to a description of the fundamental groupoids of the nerve and of the classifying space of the cover {gV : g ∈ G} of X. There is a subtle question of the description... |

3 |
Some non-abelian methods in homotopy theory and homological algebra’, Categorical topology
- BROWN
- 1983
(Show Context)
Citation Context ...les: The notion is due to J. H. C. Whitehead [149], the name being used in [150]. Surveys of their use and relationships with classical notions of homotopy theory and homological algebra are given in =-=[17, 18, 30, 31]-=-. The equivalence between crossed modules and cat l -groups is given as follows. Let µ : M → P be a crossed module. Let G = P ⋊M be the semi-direct product group, using the action of P on M, and let s... |

3 |
Topological groupoids I: Universal constructions
- Brown, Hardy
(Show Context)
Citation Context ...n foliation theory were developed by Haefliger [70], and then many others - see the survey [96], the bibliography [66]. This and other uses of topological groupoids are noted in the bibliographies to =-=[21, 22]-=-. Abstract groupoids were applied by Dedecker in a series of papers on non-abelian cohomology (see [47]). The situation now is that groupoids have been used in a wide variety of areas of mathematics, ... |

3 |
Subgroups of free topological groups and free products of topological groups
- BROWN, HARDY
- 1975
(Show Context)
Citation Context ... choose a retraction ˜ G → ˜ G(H), in a manner appropriate to the presentation of ˜ G, to obtain a presentation of ˜ G(H) and so of H [44, 78]. This strategy also gives results for topological groups =-=[23, 110]-=-. Another use of X⋊G is in ergodic theory. G. W. Mackey describes in [100] his route to the use of groupoids. The starting point was the question: granted that a transitive action of a group G on a se... |

3 |
Lifting amalgamated sums and other colimits of groups and topological
- BROWN, HEATH
- 1987
(Show Context)
Citation Context ...H,K)). This isomorphism is useful even when G,H,K are groups. It has a generalisation to the case of groupoids over a given groupoid, [40, 84]. An application of this generalisation is pointed out in =-=[24]-=-, as follows. Let f : A → B be an epimorphism of groups. Suppose B has a presentation B = colimλBλ as a colimit over a connected diagram. Let Aλ → A be the pullback of the canonical map Bλ → B by f. T... |

3 |
Crossed complexes and non-abelian extensions’, Category theory proceedings
- BROWN, HIGGINS
- 1981
(Show Context)
Citation Context ...les: The notion is due to J. H. C. Whitehead [149], the name being used in [150]. Surveys of their use and relationships with classical notions of homotopy theory and homological algebra are given in =-=[17, 18, 30, 31]-=-. The equivalence between crossed modules and cat l -groups is given as follows. Let µ : M → P be a crossed module. Let G = P ⋊M be the semi-direct product group, using the action of P on M, and let s... |

2 |
Kompositionsbegriff bei den quaternärer quadratischer Formen
- BRANDT, ‘Der
- 1924
(Show Context)
Citation Context ...r all x,y in Ob(G) there is an a in G such that sa = x,ta = y - such a groupoid we nowadays call connected or transitive. Brandt’s definition of groupoid arose out of his work for over thirteen years =-=[6, 7, 8, 9, 10]-=- on generalising to quaternary quadratic forms a composition of binary quadratic forms due to Gauss [63]. Brandt then saw how to use the notion of groupoid in generalising to the non-commutative case ... |

2 |
Groupoids as coefficients
- BROWN
- 1972
(Show Context)
Citation Context ...ct groupoid, and so the notation X ⋊ G, because this groupoid is a special case of the semi-direct product groupoid obtained from an action of a group, or more generally groupoid, on another groupoid =-=[16]-=-. A term suggested by Pradines is actor groupoid. Note that for this example, there is an identity (x, 1) for each x ∈ X. This construction is due to Ehresmann [52]. Thus we find that a set X, a group... |

1 |
Beiträge zur Galoisschen Theorie
- BAER
- 1928
(Show Context)
Citation Context ...e [81,91,92]. At about the same time as Brandt’s work, Loewy [98] introduced similar ‘compound groups’ to describe isomorphisms between conjugate field extensions. His ideas were developed by Baer in =-=[4]-=-. The most recent account of the use of groupoids in classical Galois theory seems to be that by Michler in [105]. We say more later on the use of groupoids in the Galois theory of rings1 . The topic ... |

1 |
Zur Komposition der quaternärer quadratischer Formen
- BRANDT
- 1912
(Show Context)
Citation Context ...r all x,y in Ob(G) there is an a in G such that sa = x,ta = y - such a groupoid we nowadays call connected or transitive. Brandt’s definition of groupoid arose out of his work for over thirteen years =-=[6, 7, 8, 9, 10]-=- on generalising to quaternary quadratic forms a composition of binary quadratic forms due to Gauss [63]. Brandt then saw how to use the notion of groupoid in generalising to the non-commutative case ... |

1 |
Die Hauptklassen in der Kompositionstheorie der quaternärer quadratischer Formen
- BRANDT
- 1925
(Show Context)
Citation Context ...r all x,y in Ob(G) there is an a in G such that sa = x,ta = y - such a groupoid we nowadays call connected or transitive. Brandt’s definition of groupoid arose out of his work for over thirteen years =-=[6, 7, 8, 9, 10]-=- on generalising to quaternary quadratic forms a composition of binary quadratic forms due to Gauss [63]. Brandt then saw how to use the notion of groupoid in generalising to the non-commutative case ... |

1 |
Über die Komponierbarkeit quaternärer quadratischer Formen
- BRANDT
- 1925
(Show Context)
Citation Context |

1 |
Über das associative Gesetz bei den Komposition der quaternärer quadratischer Formen
- BRANDT
- 1926
(Show Context)
Citation Context |

1 |
Topological groupoids II–covering morphisms and
- BROWN, DANESH-NARUlE, et al.
- 1976
(Show Context)
Citation Context ...n foliation theory were developed by Haefliger [70], and then many others - see the survey [96], the bibliography [66]. This and other uses of topological groupoids are noted in the bibliographies to =-=[21, 22]-=-. Abstract groupoids were applied by Dedecker in a series of papers on non-abelian cohomology (see [47]). The situation now is that groupoids have been used in a wide variety of areas of mathematics, ... |

1 |
Identities among re1ations
- BROWN, HUEBSCHMAN
- 1982
(Show Context)
Citation Context ...les: The notion is due to J. H. C. Whitehead [149], the name being used in [150]. Surveys of their use and relationships with classical notions of homotopy theory and homological algebra are given in =-=[17, 18, 30, 31]-=-. The equivalence between crossed modules and cat l -groups is given as follows. Let µ : M → P be a crossed module. Let G = P ⋊M be the semi-direct product group, using the action of P on M, and let s... |

1 | Exponential laws for topological categories, groupoids and groups and mapping spaces of colimits’ Cahiers Topologie Géom. Différentielle 20 - BROWN, NICKOLAS - 1979 |

1 |
sujet de I’existence, d’adjoints à droite aux foncteurs image réciproque dans la catégorie des catégories
- CONDUCHÉ, ‘Au
- 1972
(Show Context)
Citation Context ...morphism Hom(GxH,K) ∼ = Hom(G,HOM(H,K)). HOM(GxH,K) ∼ = HOM(G,HOM(H,K)). This isomorphism is useful even when G,H,K are groups. It has a generalisation to the case of groupoids over a given groupoid, =-=[40, 84]-=-. An application of this generalisation is pointed out in [24], as follows. Let f : A → B be an epimorphism of groups. Suppose B has a presentation B = colimλBλ as a colimit over a connected diagram. ... |

1 |
Une interprétation des relations d’équivalence dans un ensemble
- CROISOT
- 1948
(Show Context)
Citation Context ...Xbecomes a groupoid with s,t : R → X the two projections, and product (x,y)(y,z) = (x,z). whenever (x,y),(y,z) ∈ R. There is an identity, namely (x,x), for each x ∈ X. (This example is due to Croisot =-=[42]-=-.) A special case of this groupoid is the coarse groupoid X×X, which is obtained by taking R = X × X. This apparently banal and foolish example is found to play a key role in the theory and applicatio... |

1 |
The subgroup theorem for amalgamated free products, HNNconstructions and colimits
- CROWELL, SMYTHE
(Show Context)
Citation Context ... of the group G to a presentation of the groupoid ˜ G, and then to choose a retraction ˜ G → ˜ G(H), in a manner appropriate to the presentation of ˜ G, to obtain a presentation of ˜ G(H) and so of H =-=[44, 78]-=-. This strategy also gives results for topological groups [23, 110]. Another use of X⋊G is in ergodic theory. G. W. Mackey describes in [100] his route to the use of groupoids. The starting point was ... |