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Homotopy Theory, and Change of Base for Groupoids and Multiple Groupoids
, 1996
"... This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids. ..."
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Cited by 7 (6 self)
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This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.
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
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 : : ...
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.
on the category of small n–fold categories
"... small n–fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n–fold functor is a weak equivalence if and only if the diagonal of its n–fold nerve is a weak equivalence of simplicial sets. This is an n–fold analogue to Thomason ..."
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small n–fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n–fold functor is a weak equivalence if and only if the diagonal of its n–fold nerve is a weak equivalence of simplicial sets. This is an n–fold analogue to Thomason’s Quillen model structure on Cat. We introduce an n–fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n–fold nerve. As a consequence, we completely prove that the unit and counit of the adjunction between simplicial sets and n–fold categories are natural weak equivalences. 18D05, 18G55; 55U10, 55P99 1