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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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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
Higher Hopf formulae for homology via Galois Theory, preprint math.AT/0701815
, 2007
"... and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case ..."
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Cited by 10 (3 self)
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and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A
III A Functional Approach to General Topology
"... In this chapter we wish to present a categorical approach to fundamental concepts of General Topology, by providing a category X with an additional structure which allows us to display more directly the geometric properties of the objects of X regarded as spaces. Hence, we study topological properti ..."
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Cited by 3 (0 self)
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In this chapter we wish to present a categorical approach to fundamental concepts of General Topology, by providing a category X with an additional structure which allows us to display more directly the geometric properties of the objects of X regarded as spaces. Hence, we study topological properties for them, such as Hausdorff separation, compactness and local compactness, and we describe important topological constructions, such as the compactopen topology for function spaces and the StoneČech compactification. Of course, in a categorical setting, spaces are not investigated “directly ” in terms of their points and neighbourhoods, as in the traditional settheoretic setting; rather, one exploits the fact that the relations of points and parts inside a space become categorically special cases of the relation of the space to other objects in its category. It turns out that many stability properties and constructions are established more economically in the categorical rather than the settheoretic setting, leave alone the much greater level of generality and applicability. The idea of providing a category with some kind of topological structure is certainly
Galois Groupoids and Covering Morphisms in Topos Theory
"... The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given ..."
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Cited by 2 (2 self)
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The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given by a pair of adjoint functors, and (2) to discuss a good notion of covering morphism of a topos E over S which is general enough to include, in addition to the covering projections determined by the locally constant objects, also the unramified morphisms of topos theory given by those local homeomorphisms which are at the same time complete spreads in the sense of BungeFunk (1996, 1998).
Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
, 2009
"... A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromod ..."
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A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic JahnTeller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non–Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier–Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasitriangular, quasiHopf algebras, bialgebroids, GrassmannHopf algebras and Higher Dimensional Algebra. On the one hand, this quantum
Van Kampen theorems for toposes
"... In this paper we introduce the notion of an extensive 2category, to be thought of as a "2category of generalized spaces". We consider an extensive 2category K equipped with a binaryproductpreserving pseudofunctor C : K CAT, which we think of as specifying the "coverings" of our generalize ..."
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In this paper we introduce the notion of an extensive 2category, to be thought of as a "2category of generalized spaces". We consider an extensive 2category K equipped with a binaryproductpreserving pseudofunctor C : K CAT, which we think of as specifying the "coverings" of our generalized spaces. We prove, in this context, a van Kampen theorem which generalizes and refines one of Brown and Janelidze. The local properties required in this theorem are stated in terms of morphisms of effective descent for the pseudofunctor C . We specialize the general van Kampen theorem to the 2category Top S of toposes bounded over an elementary topos S , and to its full sub 2category LTop S determined by the locally connected toposes, after showing both of these 2categories to be extensive. We then consider three particular notions of coverings on toposes corresponding respectively to local homeomorphisms, covering projections, and unramified morphisms; in each case we deduce a suitable version of a van Kampen theorem in terms of coverings and, under further hypotheses, also one in terms of fundamental groupoids. An application is also given to knot groupoids and branched coverings. Along the way
Contents
, 2007
"... 1 Thanks must go to Katherine Butterfield without whom I would never have embarked on this, and to Ian Chivers who rekindled my interest. ..."
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1 Thanks must go to Katherine Butterfield without whom I would never have embarked on this, and to Ian Chivers who rekindled my interest.
Universal Covers and Category Theory in Polynomial and Differential Galois Theory
"... Abstract. The category of finite dimensional modules for the proalgebraic differential Galois group of the differential Galois theoretic closure of a differential field F is equivalent to the category of finite dimensional F spaces with an endomorphism extending the derivation of F. This paper prese ..."
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Abstract. The category of finite dimensional modules for the proalgebraic differential Galois group of the differential Galois theoretic closure of a differential field F is equivalent to the category of finite dimensional F spaces with an endomorphism extending the derivation of F. This paper presents an expository proof of this fact modeled on a similar equivalence from polynomial Galois theory, whose proof is also presented as motivation. 1
STRONGLY SEPARABLE MORPHISMS IN GENERAL CATEGORIES
"... Dedicated to Dominique Bourn on the occasion of his sixtieth birthday ..."