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18
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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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
Higher central extensions and Hopf formulae
, 902
"... Higher extensions and higher central extensions, which are of importance to nonabelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained. 0 ..."
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Higher extensions and higher central extensions, which are of importance to nonabelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained. 0
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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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).
Relative commutator theory in varieties of Ωgroups
 J. Pure Appl. Algebra
"... We introduce a new notion of commutator which depends on a choice of subvariety in any variety of Ωgroups. We prove that this notion encompasses Higgins’s commutator, Fröhlich’s central extensions and the Peiffer commutator of precrossed modules. Keywords: Commutator, Ωgroup, central extension, Pe ..."
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We introduce a new notion of commutator which depends on a choice of subvariety in any variety of Ωgroups. We prove that this notion encompasses Higgins’s commutator, Fröhlich’s central extensions and the Peiffer commutator of precrossed modules. Keywords: Commutator, Ωgroup, central extension, Peiffer commutator 0
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
THE THIRD COHOMOLOGY GROUP CLASSIFIES DOUBLE CENTRAL EXTENSIONS
, 2010
"... We characterise the double central extensions in a semiabelian category in terms of commutator conditions. We prove that the third cohomology group H3 (Z, A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A. ..."
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We characterise the double central extensions in a semiabelian category in terms of commutator conditions. We prove that the third cohomology group H3 (Z, A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A.
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
FUNDAMENTAL PUSHOUT TOPOSES
"... Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spir ..."
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Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zerodimensional locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption.
Extended Galois Theory And Dissonant Morphisms
"... For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of socalled weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various ..."
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For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of socalled weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various known results in categorical Galois theory and in particular two results of M. Barr and R. Diaconescu [BD]. We also give an elaborate list of examples and applications.