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35
Higher Hopf formulae for homology via Galois Theory
, 2007
"... We use Janelidze’s Categorical Galois Theory to extend Brown 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 on ..."
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Cited by 19 (15 self)
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We use Janelidze’s Categorical Galois Theory to extend Brown 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
, 2009
"... 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. ..."
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Cited by 9 (6 self)
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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.
Galois theory and double central extensions
 Homology, Homotopy Appl
"... We define a Galois structure between central extensions and extensions in a Maltsev variety. By using the theory of commutators we introduce double central extensions. We then prove that the covering morphisms relative to this Galois structure are precisely the double central extensions. ..."
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Cited by 8 (2 self)
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We define a Galois structure between central extensions and extensions in a Maltsev variety. By using the theory of commutators we introduce double central extensions. We then prove that the covering morphisms relative to this Galois structure are precisely the double central extensions.
Protoadditive functors, derived torsion theories and homology
 J. Pure Appl. Algebra
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The third cohomology group classifies double central extensions
 PréPublicações DMUC 0855 (2008
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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 3 (3 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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Cited by 3 (3 self)
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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
HIGHER CENTRAL EXTENSIONS VIA COMMUTATORS
, 2012
"... We prove that all semiabelian categories with the the Smith is Huq property satisfy the Commutator Condition (CC): higher central extensions may be characterised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commutators suffice. As a consequence, in the presence of enough p ..."
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Cited by 3 (1 self)
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We prove that all semiabelian categories with the the Smith is Huq property satisfy the Commutator Condition (CC): higher central extensions may be characterised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commutators suffice. As a consequence, in the presence of enough projectives we obtain explicit Hopf formulae for homology with coefficients in the abelianisation functor, and an interpretation of cohomology with coefficients in an abelian object in terms of equivalence classes of higher central extensions. We also give a counterexample against (CC) in the semiabelian category of (commutative) loops.
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