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Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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Cited by 18 (7 self)
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
Finite groups, spherical 2categories, and 4manifold invariants. arXiv:math.QA/9903003
"... In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], althou ..."
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Cited by 16 (5 self)
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In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the statesum invariants of Birmingham and Rakowski [11, 12, 13], who studied DijkgraafWitten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are nontrivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3types, such as [15], for example. 1 1
Homotopy types of strict 3groupoids
, 1988
"... It has been difficult to see precisely the role played by strict ncategories in the nascent theory of ncategories, particularly as related to ntruncated homotopy types of spaces. We propose to show in a fairly general setting that one cannot obtain all 3types by any reasonable realization functo ..."
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Cited by 13 (0 self)
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It has been difficult to see precisely the role played by strict ncategories in the nascent theory of ncategories, particularly as related to ntruncated homotopy types of spaces. We propose to show in a fairly general setting that one cannot obtain all 3types by any reasonable realization functor 1 from strict 3groupoids (i.e. groupoids in the sense of [20]). More precisely we show that one does not obtain the 3type of S 2. The basic reason is that the Whitehead bracket is nonzero. This phenomenon is actually wellknown, but in order to take into account the possibility of an arbitrary reasonable realization functor we have to write the argument in a particular way. We start by recalling the notion of strict ncategory. Then we look at the notion of strict ngroupoid as defined by Kapranov and Voevodsky [20]. We show that their definition is equivalent to a couple of other naturallooking definitions (one of these equivalences was left as an exercise in [20]). At the end of these first sections, we have a picture of strict 3groupoids having only one object and one 1morphism, as being equivalent to abelian monoidal objects (G, +) in the category of groupoids, such that (π0(G), +) is a group. In the case in question, this group will be π2(S 2) = Z. Then comes the main
Internal categorical structure in homotopical algebra
 Proceedings of the IMA workshop ?nCategories: Foundations and Applications?, June 2004, (to appear). CROSSED MODULES AND PEIFFER CONDITION 135 [Ped95] [Por87
, 1995
"... Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1. ..."
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Cited by 3 (2 self)
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Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1.
www.arpapress.com/Volumes/Vol9Issue2/IJRRAS_9_2_01.pdf QUANTUM SYMMETRIES, OPERATOR ALGEBRA AND QUANTUM GROUPOID REPRESENTATIONS: PARACRYSTALLINE SYSTEMS, TOPOLOGICAL ORDER, SUPERSYMMETRY AND GLOBAL SYMMETRY BREAKING
, 2011
"... Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applicati ..."
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Novel approaches to extended quantum symmetry, paracrystals, quasicrystals, noncrystalline solids, topological order, supersymmetry and spontaneous, global symmetry breaking are outlined in terms of quantum groupoid, quantum double groupoids and dual, quantum algebroid structures. Physical applications of such quantum groupoid and quantum algebroid representations to quasicrystalline structures and paracrystals, quantum gravity, as well as the applications of the Goldstone and Noether's theorems to: phase transitions in superconductors/superfluids, ferromagnets, antiferromagnets, mictomagnets, quasiparticle (nucleon) ultrahot plasmas, nuclear fusion, and the integrability of quantum systems are also considered. Both conceptual developments and novel approaches to Quantum theories are here proposed starting from existing Quantum Group Algebra (QGA), Algebraic Quantum Field Theories (AQFT), standard and effective Quantum Field Theories (QFT), as well as the refined `machinery ' of
Triadic van Kampen and Hurewicz Theorems ∗
, 2012
"... In [BH5] it is shown how the Relative Hurewicz Theorem follows from a Generalised Van Kampen Theorem (GVKT) for the fundamental crossed complex of a filtered space, and in [BL3] it is shown how a new multirelative Hurewicz Theorem follows from a GVKT for the fundamental cat ngroup of an ncube of s ..."
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In [BH5] it is shown how the Relative Hurewicz Theorem follows from a Generalised Van Kampen Theorem (GVKT) for the fundamental crossed complex of a filtered space, and in [BL3] it is shown how a new multirelative Hurewicz Theorem follows from a GVKT for the fundamental cat ngroup of an ncube of spaces. The purpose of this paper is to advertise and explain some implications and special cases of these GVKTs, and also to show how they came to be found. 1 Colimits of relative homotopy groups and the Relative Hurewicz Theorem 1 Although the GVKT is stated in [BH2,BH5] for crossed complexes (over groupoids), it is an important point that the main content of the final result ([BH5] Theorem C) can be summarised as a theorem on relative homotopy groups considered as modules or crossed modules over the fundamental group. Recall that if P is a group then a crossed Pmodule consists of a group M, an action of P on M on the left, say, written (m, p) ↦ → p m, and a morphism of groups µ: M → P satisfying the axioms: CM1) µ ( p m) = pmp −1; CM2) mnm −1 = µm n; for all m, n ∈ M, p ∈ P. For background in examples and applications of crossed modules, see