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21
Colimit Theorems for Relative Homotopy Groups
, 2008
"... This is the second of two papers whose main purpose is to prove a generalisation to all dimensions of the SeifertVan Kampen theorem on the fundamental group of a union of spaces. The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ω ..."
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Cited by 77 (36 self)
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This is the second of two papers whose main purpose is to prove a generalisation to all dimensions of the SeifertVan Kampen theorem on the fundamental group of a union of spaces. The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ωgroupoids and C of crossed complexes were defined, and the principal result
Private communication
, 2005
"... It was shown in [4] that a fibration p: E+B of groupoids gives rise to a family of exact sequence5, ax (1.1) l+F,,y(x~+E(~)+B(px) q,F’xvoE+~OB ..."
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Cited by 70 (0 self)
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It was shown in [4] that a fibration p: E+B of groupoids gives rise to a family of exact sequence5, ax (1.1) l+F,,y(x~+E(~)+B(px) q,F’xvoE+~OB
From groups to groupoids: a brief survey
 Bull. London Math. Soc
, 1987
"... 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 o ..."
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Cited by 58 (7 self)
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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 grouptheoretical work, it can be useful to take a path through the world of groupoids.
Determination Of A Double Lie Groupoid By Its Core Diagram
 J. Pure Appl. Algebra
, 1992
"... In a double groupoid S, we show that there is a canonical groupoid structure on the set of those squares of S for which the two source edges are identities; we call this the core groupoid of S. The target maps from the core groupoid to the groupoids of horizontal and vertical edges of S are now base ..."
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Cited by 27 (14 self)
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In a double groupoid S, we show that there is a canonical groupoid structure on the set of those squares of S for which the two source edges are identities; we call this the core groupoid of S. The target maps from the core groupoid to the groupoids of horizontal and vertical edges of S are now basepreserving morphisms whose kernels commute, and we call the diagram consisting of the core groupoid and these two morphisms the core diagram of S. If S is a double Lie groupoid, and each groupoid structure on S satisfies a natural double form of local triviality, we show that the core diagram determines S and, conversely, that a locally trivial double Lie groupoid may be constructed from an abstractly given core diagram satisfying some natural additional conditions. In the algebraic case, the corresponding result includes the known equivalences between crossed modules, special double groupoids with special connection (Brown and Spencer), and cat 1 groups (Loday). These cases correspon...
Covering groups of nonconnected topological groups, and the monodromy groupoid of a topological group
, 1993
"... All spaces are assumed to be locally path connected and semilocally 1connected. Let X be a connected topological group with identity e, and let p: ˜ X → X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for a ..."
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Cited by 13 (10 self)
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All spaces are assumed to be locally path connected and semilocally 1connected. Let X be a connected topological group with identity e, and let p: ˜ X → X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ˜e in ˜ X with p˜e = e there is a unique structure of topological group on ˜ X such that ˜e is the
Free crossed resolutions of groups and presentations of modules of identities among relations
, 2008
"... ..."
Differential operators and actions of Lie algebroids
 in Quantization, Poisson Brackets and Beyond
"... Abstract. We demonstrate that the notions of derivative representation of a Lie algebra on a vector bundle, of semilinear representation of a Lie group on a vector bundle, and related concepts, may be understood in terms of representations of Lie algebroids and Lie groupoids, and we indicate how th ..."
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Cited by 9 (1 self)
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Abstract. We demonstrate that the notions of derivative representation of a Lie algebra on a vector bundle, of semilinear representation of a Lie group on a vector bundle, and related concepts, may be understood in terms of representations of Lie algebroids and Lie groupoids, and we indicate how these notions extend to derivative representations of Lie algebroids and semilinear representations of Lie groupoids in general.
The Fundamental Groupoid as a Topological Groupoid
 Proc. Edinburgh Math. Soc
, 1975
"... Let X be a topological space. Then we may define the fundamental groupoid nX and also the quotient groupoid (nX)/N for N any wide, totally disconnected, normal subgroupoid N of nX (1). The purpose of this note is to show that if X is locally pathconnected and semilocally 1connected, then the topo ..."
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Cited by 7 (4 self)
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Let X be a topological space. Then we may define the fundamental groupoid nX and also the quotient groupoid (nX)/N for N any wide, totally disconnected, normal subgroupoid N of nX (1). The purpose of this note is to show that if X is locally pathconnected and semilocally 1connected, then the topology of X
Galois Theory of Second Order Covering Maps of Simplicial Sets
 J. Pure Appl. Algebra
, 1995
"... this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but ..."
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Cited by 6 (3 self)
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this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but
On (co)morphisms of Lie pseudoalgebras and groupoids
 J. Algebra
"... We give a unified description of morphisms and comorphisms of Lie pseudoalgebras, showing that the both types of morphisms can be regarded as subalgebras of a Lie pseudoalgebra, called the ψsum. We also provide similar descriptions for morphisms and comorphisms of Lie algebroids and groupoids. 1 ..."
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Cited by 5 (1 self)
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We give a unified description of morphisms and comorphisms of Lie pseudoalgebras, showing that the both types of morphisms can be regarded as subalgebras of a Lie pseudoalgebra, called the ψsum. We also provide similar descriptions for morphisms and comorphisms of Lie algebroids and groupoids. 1