## Fibrations of groupoids (1970)

Venue: | J. Algebra |

Citations: | 24 - 15 self |

### BibTeX

@ARTICLE{Brown70fibrationsof,

author = {Ronald Brown},

title = {Fibrations of groupoids},

journal = {J. Algebra},

year = {1970},

pages = {103--132}

}

### Years of Citing Articles

### OpenURL

### Abstract

theory, and change of base for groupoids and multiple

### Citations

133 |
Review of the elements of 2-categories
- Kelly, Street
(Show Context)
Citation Context ... homotopy addition lemma [9]. They also have a monoidal closed structure, related to a notion of homotopy, and which may be derived from that for the case of all dimensions given in [14]. 2-groupoids =-=[45, 12]-=- These are nearer to the well used 2-categories. On the other hand, their monoidal closed structure, which follows from the equivalence with the previous example, seems more difficult to describe than... |

122 | On the algebra of cubes - Brown, Higgins - 1981 |

82 |
Formal Category Theory: Adjointness for 2-categories
- Gray
- 1974
(Show Context)
Citation Context ...which follows from the equivalence with the previous example, seems more difficult to describe than in either of the previous examples. The corresponding case of 2-categories is dealt with by Gray in =-=[32, 33]-=-. The equivalence of 2-categories with a form of double categories with connection is due to Spencer [53, 54], but seems not to have been otherwise exploited, except in recent work of Verity on cubica... |

75 | Colimit theorems for relative homotopy groups
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...crete groupoid, with only identities. This category is equivalent to at least three other categories, of which two are those of crossed modules over groupoids, and of double groupoids with connection =-=[11]-=-. (The case of these equivalences when P is a singleton is due to Brown and Spencer [19, 20].) Each of these forms of double groupoids have their own particular value, and circumstances when they are ... |

59 | From groups to groupoids: a briefsurvey
- Brown
- 1987
(Show Context)
Citation Context ...scheme, and so on, especially as groupoids can carry a wider range of non trivial structures than can groups. For a general survey of groupoids, with 160 references, including many omitted above, see =-=[3]-=-. It is also interesting to note the relation of ordered groupoids to inverse semigroups, and so to notions of partial symmetry. In this area, Lawson has developed some of Ehresmann’s foundational wor... |

54 |
Topology: A Geometric Account of General Topology
- Brown
- 1988
(Show Context)
Citation Context ...ble way towards obtaining new information was conjectured in 1967 in [2], suggested by the new use for computations of the fundamental groupoid. The successes of the fundamental groupoid, as shown in =-=[2, 4]-=-, seemed to stem from the fact that it had structure in two dimensions, 0 and 1. This raised the prospect of reflecting more of the way homotopy theory worked by using algebraic objects with structure... |

50 | On the connection between the second relative homotopy groups of some related spaces
- Brown, Higgins
- 1978
(Show Context)
Citation Context ... with connection [19, 10] These are an essential tool in one proof of the Van Kampen Theorem for the fundamental crossed module, because they nicely handle subdivision and the homotopy addition lemma =-=[9]-=-. They also have a monoidal closed structure, related to a notion of homotopy, and which may be derived from that for the case of all dimensions given in [14]. 2-groupoids [45, 12] These are nearer to... |

42 |
Catégories et structures
- Ehresmann
- 1965
(Show Context)
Citation Context ...f giving a model for describing “algebraic inverses to subdivision”. A double groupoid is a set with two groupoid structures each of which is a morphism for the other. This notion is due to Ehresmann =-=[27]-=-. It can also be thought of as comprising an underlying diagram of sets S V H such that H and V are groupoids over P, and there are two groupoid structures on S, one over H, one over V , and all satis... |

32 |
Higher Dimensional Crossed Modules and the Homotopy
- Ellis, Steiner
- 1987
(Show Context)
Citation Context ...equivalence of homotopy categories. This shows that multiple groupoids are complicated objects, since homotopy theory is known to be complicated. In fact catn-groups are shown by Ellis and Steiner in =-=[31]-=- to be equivalent to what they call crossed n-cubes of groups. It is intriguing that the theory of the latter objects includes a lot of commutator theory, so that once again multiple groupoids link wi... |

30 |
Identities among relations
- Brown, Huebschmann
- 1982
(Show Context)
Citation Context ...standard second relative homotopy groups of homotopy theory, and with a long tradition of homological algebra, including chains of syzygies, and, for the non abelian cases, identities among relations =-=[15]-=-. Such links seem non accidental. Now let us go back to the double groupoid of diagram (2), a more general structure than those just considered. The idea is that the groupoid structures associated to ... |

27 | Groupoids and Van Kampen’s theorem
- Brown
- 1967
(Show Context)
Citation Context ...in which only neighbouring dimensions can interact. Thus any new methods of obtaining some new information are of interest. A possible way towards obtaining new information was conjectured in 1967 in =-=[2]-=-, suggested by the new use for computations of the fundamental groupoid. The successes of the fundamental groupoid, as shown in [2, 4], seemed to stem from the fact that it had structure in two dimens... |

27 | Determination of a double Lie groupoid by its core diagram
- Brown, Mackenzie
- 1992
(Show Context)
Citation Context ...mathematics, like other combinations of structures, for example � P (2) 181rings. However, some types of double groupoids may be described in other terms. The most general known results are given in =-=[18]-=-. The category of double groupoids contains a category of 2-groupoids, namely the double groupoids as above in which the groupoid V over P is the discrete groupoid, with only identities. This category... |

25 |
Twodimensional homotopy and combinatorial group theory
- Hog-Angeloni, Metzler, et al.
(Show Context)
Citation Context ...inning of a non abelian free crossed resolution of Coker ι, and the kernel of the boundary ∂ gives the identities among the relations. There is considerable study of this notion in the case of groups =-=[15, 43]-=-. Other presentations of induced crossed modules are given in [9]. In particular, if ι : P → Q is an epimorphism, and µ : M → P is a crossed module, then ι∗M is the quotient of M by the group [M, P ] ... |

22 |
Kampen theorems for diagrams of spaces
- BROWN, LoDAY, et al.
- 1987
(Show Context)
Citation Context ...an extensive use in algebraic topology, homological algebra, and algebraic K-theory, which gives an advantage to this category. It is crucial (at least for the group case) in the use of cat n -groups =-=[49, 16]-=-. The notion of homotopy has been written down in this case [51], but not the monoidal closed structure. One of the comforting features of work on double and multiple groupoids has been the links with... |

22 |
G -groupoids, crossed modules and the fundamental groupoid of a topological group
- Brown, Spencer
(Show Context)
Citation Context ...r categories, of which two are those of crossed modules over groupoids, and of double groupoids with connection [11]. (The case of these equivalences when P is a singleton is due to Brown and Spencer =-=[19, 20]-=-.) Each of these forms of double groupoids have their own particular value, and circumstances when they are most appropriate. Crossed modules over groupoids [10] These objects are more obviously relat... |

21 | Finite induced crossed modules and the homotopy 2-type of mapping cones
- Brown, Wensley
(Show Context)
Citation Context ...∈ M, k ∈ Ker ι. So the interest is in the case when P is a subgroup of Q. Motivated by the applications, Chris Wensley and I have given recently a range of new calculations of induced crossed modules =-=[21, 22, 23]-=-. One of our results is that if M and the index of P in Q are finite, then the induced crossed module ι∗M is finite [21]. This gives further point to finding explicit calculations. One aim described i... |

20 | Crossed complexes and chain complexes with operators
- Brown, Higgins
- 1990
(Show Context)
Citation Context ...iv) chain complexes [13]. For these reasons, it is natural to attempt to compute with these objects rather than with the other forms. There is a useful monoidal closed structure on this category, see =-=[14]-=-. Double groupoids with connection [19, 10] These are an essential tool in one proof of the Van Kampen Theorem for the fundamental crossed module, because they nicely handle subdivision and the homoto... |

18 | Homotopical excision, and Hurewicz theorems for n-cubes of spaces
- Brown, Loday
- 1987
(Show Context)
Citation Context ...s stimulated a lot of other work, including other proofs of the Hopf formula. However the special case of even the triadic Hurewicz theorem, described in [5], has no other proof to date. Results from =-=[17]-=- on induced crossed squares are used by Ellis in [30] to define what he calls “free crossed squares” and to give applications to 3-dimensional combinatorial homotopy theory. Thus applications of multi... |

17 | Computing crossed modules induced by an inclusion of a normal subgroup, with applications to homotopy 2-types
- Brown, Wensley
- 1995
(Show Context)
Citation Context ...∈ M, k ∈ Ker ι. So the interest is in the case when P is a subgroup of Q. Motivated by the applications, Chris Wensley and I have given recently a range of new calculations of induced crossed modules =-=[21, 22, 23]-=-. One of our results is that if M and the index of P in Q are finite, then the induced crossed module ι∗M is finite [21]. This gives further point to finding explicit calculations. One aim described i... |

15 |
Éléments de mathématique. Livre II: Algèbre. Chap. 8: Modules et anneaux semisimples. (French) Actualités scientifiques et industrielles
- Bourbaki
- 1958
(Show Context)
Citation Context ...m Brandt’s attempts to extend to quaternary forms Gauss’ work on the composition of binary quadratic forms, which has a strong place in Disquitiones Arithmeticae. It is of interest here that Bourbaki =-=[1]-=-, p.153, cites this composition as an influential early example of a composition law which arose not from numbers, even taken in a broad sense, but from distant analogues1 . Brandt found that 1 C’est ... |

14 |
Höherdimensionale Homotopiegruppen
- Čech
- 1932
(Show Context)
Citation Context ...roups. This shows that the higher homotopy groups do not generalise the fundamental group, a fact which led to an initial disappointment with Čech’s proposal of these groups at the 1932 ICM in Zurich =-=[24]-=-. Later work on homotopy groups continues to move away from group theoretic methods. It is worth explaining how groupoids arose. The notion of groupoid dates from Brandt’s attempts to extend to quater... |

14 |
Presentations of groupoids, with applications to groups
- Higgins
- 1964
(Show Context)
Citation Context ...epresentation theory. We discuss briefly right adjoints to pullbacks, which do not always exist in algebraic categories, in the penultimate section. 1 Change of base for groupoids Higgins’ 1964 paper =-=[38]-=- applied groupoids to group theory, independently of earlier work of Hasse [35]. One main tool was the notion of covering morphism of groupoids, motivated by the notion of covering map of spaces. The ... |

13 |
Double groupoids and crossed
- Brown, Spencer
- 1976
(Show Context)
Citation Context ... homotopy addition lemma [9]. They also have a monoidal closed structure, related to a notion of homotopy, and which may be derived from that for the case of all dimensions given in [14]. 2-groupoids =-=[45, 12]-=- These are nearer to the well used 2-categories. On the other hand, their monoidal closed structure, which follows from the equivalence with the previous example, seems more difficult to describe than... |

12 |
Pullback functors and crossed complexes’. Cahiers Topologie Géom
- Howie
- 1979
(Show Context)
Citation Context ...tegories, this right adjoint was considered by F. Conduché in [25]. He concludes that the right adjoint exists if f is a fibration. Following this lead, an analogous result was proved by Jim Howie in =-=[44]-=- for crossed complexes, and in particular for groupoids. It follows if f : G → H is a fibration of groupoids, then the pullback functor f ∗ preserves colimits. For more on fibrations of groupoids, see... |

11 |
Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson bundles
- Higgins, Mackenzie
- 1993
(Show Context)
Citation Context ...ample with induced representations, and with change of base for slice categories, i.e. categories C/A where A is an object of the category C and the slice category has objects the morphisms to A (see =-=[50, 42]-=-). The algebraic structures we consider are forms of multiple groupoid, and the corresponding “higher homotopy groupoids” give generalisations of the fundamental group. Thus the following extensions s... |

10 | Hopf formulae for the higher homology of a group
- Brown, Ellis
- 1988
(Show Context)
Citation Context ...ins, and so gave a kind of culminating point of the route from the Van Kampen Theorem described in section 1, for the fundamental groupoid. This general Hurewicz theorem is used by Brown and Ellis in =-=[6]-=- to give a Hopf formula for the (n + 1)-st homology of a group G, in terms of n normal subgroups R1, R2, . . . , Rn of a free group F such that F/(R1 . . . Rn) is isomorphic to G and various other quo... |

10 |
Double groupoids and crossed modules’, Cah
- Brown
- 1976
(Show Context)
Citation Context ...r categories, of which two are those of crossed modules over groupoids, and of double groupoids with connection [11]. (The case of these equivalences when P is a singleton is due to Brown and Spencer =-=[19, 20]-=-.) Each of these forms of double groupoids have their own particular value, and circumstances when they are most appropriate. Crossed modules over groupoids [10] These objects are more obviously relat... |

8 |
The non-Abelian tensor product of finite groups is finite
- Ellis
- 1987
(Show Context)
Citation Context ...f M. All these yield examples of finite crossed modules. Other finite examples may be constructed from those above, the induced crossed modules described below, and coproducts [4] and tensor products =-=[16, 29]-=- of crossed P -modules. We discuss later the important free crossed modules. The geometric example of a crossed module is the boundary map ∂ : π2(X, A) → π1(A) of the second relative homotopy group of... |

4 |
Triadic Van Kampen theorems and Hurewicz theorem' , Algebraic Topology
- Brown
- 1989
(Show Context)
Citation Context ... other quotients of F are free. This result has stimulated a lot of other work, including other proofs of the Hopf formula. However the special case of even the triadic Hurewicz theorem, described in =-=[5]-=-, has no other proof to date. Results from [17] on induced crossed squares are used by Ellis in [30] to define what he calls “free crossed squares” and to give applications to 3-dimensional combinator... |

3 |
Subgroups of free topological groups and free products of topological groups
- BROWN, HARDY
- 1975
(Show Context)
Citation Context ...inal groupoid, classical subgroup theorems may be obtained by this method, in some cases giving stronger versions [39]. These methods also allow for topological versions of the main subgroup theorems =-=[7]-=-. Section 6 refers to other methods of proving the required lifting of universal morphisms without using the solution of the word problem. A major reason for passing from the fundamental group to the ... |

3 |
Lifting amalgamated sums and other colimits of groups and topological
- BROWN, HEATH
- 1987
(Show Context)
Citation Context ...sult described earlier, that the pullback of a universal morphisms of groupoids by a covering morphism is again universal, follows without requiring a solution of the word problem. Brown and Heath in =-=[8]-=- observe that an epimorphism of groups is a fibration of groupoids, and deduce that a pullback of a colimit of a connected diagram of groups by an epimorphism of groups is also a colimit of the pullba... |

3 |
Induced crossed modules and computational group theory
- Brown, Wensley
- 1995
(Show Context)
Citation Context ...∈ M, k ∈ Ker ι. So the interest is in the case when P is a subgroup of Q. Motivated by the applications, Chris Wensley and I have given recently a range of new calculations of induced crossed modules =-=[21, 22, 23]-=-. One of our results is that if M and the index of P in Q are finite, then the induced crossed module ι∗M is finite [21]. This gives further point to finding explicit calculations. One aim described i... |

3 |
Crossed squares and combinatorial
- Ellis
- 1993
(Show Context)
Citation Context ...ofs of the Hopf formula. However the special case of even the triadic Hurewicz theorem, described in [5], has no other proof to date. Results from [17] on induced crossed squares are used by Ellis in =-=[30]-=- to define what he calls “free crossed squares” and to give applications to 3-dimensional combinatorial homotopy theory. Thus applications of multiple groupoids reach parts of homotopy theory not acce... |

3 |
private communication
- Grothendieck
- 1985
(Show Context)
Citation Context ... isomorphisms of a family of structures. The fundamental groupoid of a space was well known by the 1950’s. The fundamental groupoid on a set of base points is used in [2]. Grothendieck writes in 1985 =-=[34]-=-: The idea of making systematic use of groupoids (notably fundamental groupoids of spaces, based on a given set of base points), however evident as it may look today, is to be seen as a significant co... |

3 |
Grushko’s theorem
- Higgins
- 1966
(Show Context)
Citation Context ...oid, may be derived. Since these vertex groups are isomorphic to subgroups of the original groupoid, classical subgroup theorems may be obtained by this method, in some cases giving stronger versions =-=[39]-=-. These methods also allow for topological versions of the main subgroup theorems [7]. Section 6 refers to other methods of proving the required lifting of universal morphisms without using the soluti... |

2 |
sujet d'adjoints `a droite aux foncteurs image r'eciproque dans la cat'egorie des cat'egories
- Conduch'e, `Au
- 1972
(Show Context)
Citation Context ... when f : X → Y is a morphism of the category C, and CY denotes the slice category of objects of C over Y . For C the category of small categories, this right adjoint was considered by F. Conduché in =-=[25]-=-. He concludes that the right adjoint exists if f is a fibration. Following this lead, an analogous result was proved by Jim Howie in [44] for crossed complexes, and in particular for groupoids. It fo... |

2 |
Non commutative geometry, IHES
- Connes
- 1993
(Show Context)
Citation Context ...key’s work stimulated Connes’ research on non commutative integration, and the convolution algebras of groupoids continue to be a main tool in Connes’ work on non commutative geometry. Connes remarks =-=[26]-=- that Heisenberg discovered quantum mechanics by considering the algebra of observables for the groupoid of quantum transitions, rather than the traditional group of symmetry. Here, if I is a set, the... |

2 |
Oeuvres compl`etes et comment'ees, ed Andr'ee
- Ehresmann
- 1980
(Show Context)
Citation Context ...sly or smoothly through a path and returning to the initial position but not the initial value. Groupoids form a central feature of Ehresmann’s foundational work on differential geometry and topology =-=[28]-=-. In the 1960s, G W Mackey found groupoids useful for work in ergodic theory, to give an analogue for an ergodic action of the way a group of stability of a transitive action determines that action up... |

2 |
Einige Bemerkungen uber Graphen, Kategorien und Gruppoide
- Hasse
- 1960
(Show Context)
Citation Context ...not always exist in algebraic categories, in the penultimate section. 1 Change of base for groupoids Higgins’ 1964 paper [38] applied groupoids to group theory, independently of earlier work of Hasse =-=[35]-=-. One main tool was the notion of covering morphism of groupoids, motivated by the notion of covering map of spaces. The theory of covering maps may be summarised [4] by saying that for spaces with go... |

2 |
Lifting colimits of (topological) groupoids and (topological) categories', Proceedings of the conference "Categorical Topology and its Relation to Analysis Algebra and Combinatorics
- Heath, Kamps
- 1988
(Show Context)
Citation Context ... whose colimits are preserved by the inclusion of categories (groups) ⊂ (groupoids). The construction of these right adjoints, and the applications to pullbacks of colimits, are carefully analysed in =-=[36]-=-. Some of the colimit results are shown in [37] to be derivable by other means. The fact that a covering crossed complex of a free crossed complex is again free may be proved by these methods. This re... |

2 |
Lifting colimits in various categories
- Heath, Parmenter
(Show Context)
Citation Context ...of categories (groups) ⊂ (groupoids). The construction of these right adjoints, and the applications to pullbacks of colimits, are carefully analysed in [36]. Some of the colimit results are shown in =-=[37]-=- to be derivable by other means. The fact that a covering crossed complex of a free crossed complex is again free may be proved by these methods. This result has planned uses in the context of free cr... |

2 |
Fibrations and quotients of differentiable mappings
- Higgins, Mackenzie
- 1990
(Show Context)
Citation Context ...for crossed complexes, and in particular for groupoids. It follows if f : G → H is a fibration of groupoids, then the pullback functor f ∗ preserves colimits. For more on fibrations of groupoids, see =-=[41]-=-. Philip Higgins has observed that, since a covering morphism of groupoids is a special case of a fibration, the result described earlier, that the pullback of a universal morphisms of groupoids by a ... |