Fibrations of Groupoids

doi:10.1016/0021-8693(70)90089-X

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Fibrations of Groupoids

Journal Article•DOI•

Fibrations of Groupoids

01 May 1970-Journal of Algebra (Academic Press)-Vol. 15, Iss: 1, pp 103-132

TL;DR: The notion of a covering morphism of groupoids has been developed by P. J. Higgins [4, 51 and shown to be a convenient tool in algebra, even for purely group theoretic results.

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About: This article is published in Journal of Algebra.The article was published on 1970-05-01 and is currently open access. It has received 100 citations till now. The article focuses on the topics: Cohomology & Galois cohomology.

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Journal Article•DOI•

Algebraic constructions in the category of lie algebroids

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Philip J. Higgins¹, Kirill C. H. Mackenzie¹•Institutions (1)

Durham University¹

15 Feb 1990-Journal of Algebra

TL;DR: In this article, it was shown that the Lie functor from the category of all differentiable groupoids (over arbitrary bases) and arbitrary smooth morphisms, to the class of all Lie algebroids, preserves the basic algebraic constructions known to be possible in (differentiable) groupoids.

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354 citations

Cites background from "Fibrations of Groupoids"

...Actions of (set-theoretic) groupoids on groupoids and the resulting semidirect products were studied in Brown [3, 41, following Frohlich; we briefly recall the smooth version from [ 151....
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...The term “fibration” comes from Brown [3], who introduced it for the corresponding concept for set-theoretic groupoids, and from conversations between the secondnamed author and J....
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...The term “fibration” comes from Brown [3], who introduced it for the corresponding concept for set-theoretic groupoids, and from conversations between the secondnamed author and J. Pradines, who calls such maps s-exactors....
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Journal Article•DOI•

From Groups to Groupoids: a Brief Survey

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Ronald Brown

01 Mar 1987-Bulletin of The London Mathematical Society

TL;DR: A groupoid is a small category in which every morphism is an isomorphism as mentioned in this paper, and a groupoid can be thought of as a group with many objects, or with many identities.

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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. A succinct definition is that a groupoid G is a small category in which every morphism is an isomorphism. Thus G has a set of morphisms, which we shall call just elements of G, a set Ob(G) of objects or vertices, together with functions s, t : G→ Ob(G), i : Ob(G) → G such that si = ti = 1. The functions s, t are sometimes called the source and target maps respectively. If a,b ∈ G and ta = sb, then a product or composite ab exists such that s(ab) = sa, t(ab) = tb. Further, this product is associative; the elements ix, x ∈ Ob(J), act as identities; and each element a has an inverse a with s(a) = ta, t(a) = sa,aa = isa,aaa = ita. An element a is often written as an arrow a : sa→ ta.

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310 citations

Cites background from "Fibrations of Groupoids"

...Another ‘technical point’ of course is the basic fact (and the wealth of intuitions accompanying it) that the Teichmüller groups are fundamental groups indeed,-a fact ignored it seems by most geometers, because the natural ‘spaces’ they are fundamental groups of are not topological spaces, but the…...
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...If G is a groupoid, and x, y ∈ Ob(G), then we write G(x,y) for the set of elements a in G with sa = x, ta = y, and we write G(x) for G(x, x)....
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...It is this kind of viewpoint, emphasising the algebra we know rather than that which might evolve, which perhaps has led people to fail to see properly the advantages of an algebra which models the geometry more appropriately than the usual algebra of groups....
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Book•

Topology and Groupoids

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Ronald Brown

24 Feb 2006

TL;DR: Camarena as discussed by the authors pointed out a gap in the proofs in [BT&G, Bro06] of a condition for the Phragmen-Brouwer Property not to hold; this note gives the correction in terms of a result on a pushout of groupoids.

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Abstract: Omar Antoĺın Camarena pointed out a gap in the proofs in [BT&G, Bro06] of a condition for the Phragmen–Brouwer Property not to hold; this note gives the correction in terms of a result on a pushout of groupoids, and some additional background.

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269 citations

Journal Article•DOI•

Colimit theorems for relative homotopy groups

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Ronald Brown, Philip J. Higgins¹•Institutions (1)

Durham University¹

01 Aug 1981-Journal of Pure and Applied Algebra

TL;DR: In this paper it was proved that an ω-groupoid is a special kind of Kan cubical complex, in that every box has a unique thin filler, and a version of the homotopy addition lemma, and properties of thin elements, in a groupoid were established.

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149 citations

Book•

Topology: A Geometric Account of General Topology Homotopy Types and the Fundamental Groupoid

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Ronald Brown

01 Feb 1989

148 citations

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References

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Book•

Elements of Modern Topology

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Ronald Brown

01 Jan 1968

153 citations

Book Chapter•DOI•

Fibred and Cofibred Categories

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John W. Gray

01 Jan 1966

TL;DR: The authors obtained a copy of handwritten notes from a seminar given by Chevalley at Berkeley in 1962, which treated these questions from a slightly different point of view, and discussed the Chevalleys condition.

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Abstract: Fibred categories were introduced by Gkothendieck in [SGA] and [BB190]. As far as I know these are the only easily available references to the subject. Through sheer luck, during the final preparation of this paper I obtained a copy of handwritten notes [BN] of a seminar given by Chevalley at Berkeley in 1962 which treated these questions from a slightly different point of view. We discuss the “Chevalley condition” in 3.11.

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148 citations

Book•

Proceedings of the Conference on Categorical Algebra, La Jolla, 1965

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Samuel Eilenberg

01 Jan 1966

45 citations

Journal Article•DOI•

Coglueing homotopy equivalences

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Ronald Brown¹, Philip R. Heath²•Institutions (2)

University of Hull¹, Memorial University of Newfoundland²

01 Aug 1970-Mathematische Zeitschrift

TL;DR: In this paper, the authors give conditions on the front and back squares which ensure that if q~l, q~2 and ~o are homotopy equivalences, then so also is ~.

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Abstract: in which the front square is a pull-back. Then P is often called the fibre-product o f f and p, and it is also said that ~: P ~ X is induced by f from p. The map ~: Q~ P is determined by ~01 and q)2. Our object is to give conditions on the front and back squares which ensure that if q~l, q~2 and ~o are homotopy equivalences, then so also is ~. First of all we shall assume throughout that the back square of (1.1) as well as the front square, is a pull-back. Second recall that a map q: E --* B has the W C H P (weak covering homotopy property) if it has the covering property for all homotopies Z x I ---, B which are stationary on Z x [0, 89 This property has been shown by Dold [3] and Weinzweig [7] to be convenient for studying fiber homotopy equivalences, and our results will extend some of theirs. For the rest of this section we will assume that in (1.1) p and q have the WCHP. Then our main object is the following theorem which will be proved in Sections 2 to 5.

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29 citations