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 Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A

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 Bunge-Funk (1996, 1998).

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 Jahn-Teller 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, Grassmann-Hopf algebras and Higher Dimensional Algebra. On the one hand, this quantum

In this paper we introduce the notion of an extensive 2-category, to be thought of as a "2-category of generalized spaces". We consider an extensive 2-category K equipped with a binary-product-preserving 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 2-category Top S of toposes bounded over an elementary topos S , and to its full sub 2-category LTop S determined by the locally connected toposes, after showing both of these 2-categories 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

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 zero-dimensional 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.

For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of so-called 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.

We characterise the double central extensions in a semi-abelian 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.

Abstract not found

Galois theory and a new homotopy double groupoid of a map of spaces

(communicated by George Janelidze) We show that the class of separable morphisms in the sense of G. Janelidze and W. Tholen in the case of Galois structure