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Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
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"... Abstract. This paper is the second in a series exploring the properties of a functorwhich 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 isa suitable tool for nonabelian, 2dimensional, localt ..."
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Abstract. This paper is the second in a series exploring the properties of a functorwhich 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 isa suitable tool for nonabelian, 2dimensional, localtoglobal problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for otherhigher 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 alldimensions by Philip Higgins.
Some Problems In NonAbelian Homotopical And Homological Algebra
, 1999
"... this paper is to convey some impression of the extent of an area of nonAbelian homotopical and homological algebra, by giving some of the problems, of varying degrees of diculty and of precision, which I have come across over the years and in which progress would be desirable. ..."
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this paper is to convey some impression of the extent of an area of nonAbelian homotopical and homological algebra, by giving some of the problems, of varying degrees of diculty and of precision, which I have come across over the years and in which progress would be desirable.
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
Double categories, 2categories . . .
 THEORY APPL. CATEG
, 1999
"... The main result is that two possible structures which may be imposed on an edge symmetric double category, namely a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the category of small 2categories is equivalent to the category of edg ..."
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The main result is that two possible structures which may be imposed on an edge symmetric double category, namely a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the category of small 2categories is equivalent to the category of edge symmetric double categories with thin structure.
Pasting Presentations for OmegaCategories
, 1995
"... The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !ca ..."
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The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !categories in terms of generating pasting schemes and relations between generated pastings, i.e., with pasting presentations. In this chapter I develop the necessary machinery for this. The main result, that the !category generated by a pasting presentation is universal with respect to respectable families of realizations, is a generalization of the pasting theorem. Contents 1 Introduction 3 2 Pasting schemes according to Johnson 4 2.1 Graded sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 !categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Pasting schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.4 The pasting theorem : : ...
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, 2002
"... Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional localtoglobal problems ..."
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Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional localtoglobal problems