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45
Nonabelian Algebraic Topology
, 2004
"... This is an extended account of a short presentation with this title given at the Minneapolis IMA Workshop on ‘ncategories: foundations and applications’, June 718, 2004, ..."
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This is an extended account of a short presentation with this title given at the Minneapolis IMA Workshop on ‘ncategories: foundations and applications’, June 718, 2004,
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
Crossed complexes, and free crossed resolutions for amalgamated sums and HNNextensions of groups
 Georgian Math. J
, 1999
"... Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the p ..."
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Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating nonabelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNNextensions of groups, and so obtain computations of higher homotopical syzygies in these cases. 1
Formal Homotopy Quantum Field Theories
 II : Simplicial Formal Maps
"... Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQ ..."
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Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G,1) and also when it is simply connected. Here we study general HQFTs with d = 1 and target a general 2type, giving a common generalisation of the classifying algebraic structures for the two cases previously known. The algebraic models for 2types that we use are crossed modules, C, and we introduce a notion of formal Cmap, which extends the usual latticetype constructions to this setting. This leads to a classification of ‘formal ’ 2dimensional HQFTs with target C,
Higher cospans and weak cubical categories (Cospans in Algebraic Topology
 I), Theory Appl. Categ
"... form a cubical set with compositions x +i y in all directions, which are computed using pushouts and behave ‘categorically ’ in a weak sense, up to suitable comparisons. Actually, we work with a ‘symmetric cubical structure’, which includes the transposition symmetries, because this allows for a str ..."
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form a cubical set with compositions x +i y in all directions, which are computed using pushouts and behave ‘categorically ’ in a weak sense, up to suitable comparisons. Actually, we work with a ‘symmetric cubical structure’, which includes the transposition symmetries, because this allows for a strong simplification of the coherence conditions. These notions will be used in subsequent papers to study topological cospans and their use in Algebraic Topology, from tangles to cobordisms of manifolds. We also introduce the more general notion of a multiple category, where to start witharrows belong to different sorts, varying in a countable family, and symmetries must be dropped. The present examples seem to show that the symmetric cubical case is better suited for topological applications.
The Persistent Abstract
 Machine”, Proceedings of the Third International Workshop on Persistent Object Systems
, 1989
"... The notion of local equivalence relation on a topological space is generalised to that of local subgroupoid. The main result is the construction of the holonomy and monodromy groupoids of certain Lie local subgroupoids, and the formulation of a monodromy principle on the extendibility of local Lie m ..."
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The notion of local equivalence relation on a topological space is generalised to that of local subgroupoid. The main result is the construction of the holonomy and monodromy groupoids of certain Lie local subgroupoids, and the formulation of a monodromy principle on the extendibility of local Lie morphisms.
Homotopies and automorphism of crossed modules of groupoids
, 2003
"... Abstract. We give a detailed description of the structure of the actor 2crossed module related to the automorphisms of a crossed module of groupoids. This generalises work of Brown and Gilbert for the case of crossed modules of groups, and part of this is needed for work on 2dimensional holonomy t ..."
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Abstract. We give a detailed description of the structure of the actor 2crossed module related to the automorphisms of a crossed module of groupoids. This generalises work of Brown and Gilbert for the case of crossed modules of groups, and part of this is needed for work on 2dimensional holonomy to be developed elsewhere.
Bisimulation for higherdimensional automata. A geometric interpretation. Research report R200501
 Department of Mathematical Sciences, Aalborg University
, 2005
"... We show how parallel composition of higherdimensional automata (HDA) can be expressed categorically in the spirit of Winskel & Nielsen. Employing the notion of computation path introduced by van Glabbeek, we define a new notion of bisimulation of HDA using open maps. We derive a connection betw ..."
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We show how parallel composition of higherdimensional automata (HDA) can be expressed categorically in the spirit of Winskel & Nielsen. Employing the notion of computation path introduced by van Glabbeek, we define a new notion of bisimulation of HDA using open maps. We derive a connection between computation paths and carrier sequences of dipaths and show that bisimilarity of HDA can be decided by the use of geometric techniques.
R.Picken, A Cubical Set Approach to 2Bundles with Connection and Wilson Surfaces
"... In the context of nonabelian gerbes we define a cubical version of categorical group 2bundles with connection over a smooth manifold and consider their twodimensional parallel transport with the aim of defining nonabelian Wilson surface functionals. Key words and phrases: cubical set; nonabelia ..."
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In the context of nonabelian gerbes we define a cubical version of categorical group 2bundles with connection over a smooth manifold and consider their twodimensional parallel transport with the aim of defining nonabelian Wilson surface functionals. Key words and phrases: cubical set; nonabelian gerbe; 2bundle; twodimensional holonomy; twodimensional parallel transport; crossed module; categorical group; double groupoid; Higher Gauge Theory; Wilson surface; Wilson sphere; knotted sphere
EXACT SEQUENCES OF FIBRATIONS OF CROSSED COMPLEXES, HOMOTOPY CLASSIFICATION OF MAPS, AND NONABELIAN EXTENSIONS OF GROUPS
"... The classifying space of a crossed complex generalises the construction of EilenbergMac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy ..."
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The classifying space of a crossed complex generalises the construction of EilenbergMac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CWcomplex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory.