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51
The algebra of cubes
, 2002
"... This is the first of two papers whose main purpose is to prove a generalization of the SeifertVan Kampen theorem on the fundamental group of a union of spaces. This generalisation (Theorem C of [8]) will give information in all dimensions and will include as special cases not only the above theorem ..."
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Cited by 122 (39 self)
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This is the first of two papers whose main purpose is to prove a generalization of the SeifertVan Kampen theorem on the fundamental group of a union of spaces. This generalisation (Theorem C of [8]) will give information in all dimensions and will include as special cases not only the above theorem (without the usual assumptions of pathconnectedness) but also
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 75 (34 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
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...
Fibrations of groupoids
 J. Algebra
, 1970
"... theory, and change of base for groupoids and multiple ..."
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Cited by 24 (15 self)
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theory, and change of base for groupoids and multiple
On finite induced crossed modules, and the homotopy 2type of mapping cones
 THEORY AND APPLICATIONS OF CATEGORIES
, 1995
"... Results on the niteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of ..."
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Cited by 21 (18 self)
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Results on the niteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of some nite crossed modules are given, using crossed complex methods.
A homotopy double groupoid of a Hausdorff space II: A van Kampen Theorem
 THEORY AND APPLICATIONS OF CATEGORIES
, 2005
"... This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2dimensional, localtoglobal ..."
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Cited by 20 (11 self)
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This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2dimensional, localtoglobal problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These results have recently been generalised to all dimensions by Philip Higgins.
Homotopical excision, and Hurewicz theorems, for ncubes of spaces
 Proc. London Math. Soc
, 1987
"... The fact that the relative homotopy groups do not satisfy excision makes the computation of absolute homotopy groups difficult in comparison with homology groups. The failure of excision is measured by triad homotopy groups πn(X; A, B), with n � 3 (for n = 2, this gives a based set), which fit into ..."
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Cited by 18 (9 self)
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The fact that the relative homotopy groups do not satisfy excision makes the computation of absolute homotopy groups difficult in comparison with homology groups. The failure of excision is measured by triad homotopy groups πn(X; A, B), with n � 3 (for n = 2, this gives a based set), which fit into an exact sequence.
Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional localtoglobal problems
 in Proceedings of the Fields Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, September 2328, Fields Institute Communications,43
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