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On the representation theory of Galois and atomic topoi
- J. Pure Appl. Algebra
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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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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).
The fundamental progroupoid of a general topos
"... Abstract. It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the fundamental progroupoid, and that this progroup ..."
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Abstract. It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the fundamental progroupoid, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004. introduction. It is well known that if E is a locally connected topos then the category of covering projections (that is, locally constant objects) is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete
Van Kampen theorems for toposes
"... 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&quo ..."
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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
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 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.
Constructive Theory of Galois Toposes
"... Galois toposes were considered by Grothendieck in connection with the fundamental progroup of a topos. They were subsequently shown by Moerdijk to correspond to the classifying toposes of prodiscrete localic groups. Over an arbitrary base topos S, the (coverings) fundamental group constructed by Bun ..."
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Galois toposes were considered by Grothendieck in connection with the fundamental progroup of a topos. They were subsequently shown by Moerdijk to correspond to the classifying toposes of prodiscrete localic groups. Over an arbitrary base topos S, the (coverings) fundamental group constructed by Bunge is (the classifying topos of) a prodiscrete localic groupoid. We introduce a notion of Galois topos relative to an arbritrary base topos S and show that it reduces to the usual when the base topos is Set. The constructive theory of S-Galois toposes requires some nontrivial modi cations. Locally constant coverings must be replaced by the locally componentwise constant coverings. Whereas the former satisfy the unique pathlifting property, the latter only have the path-lifting property stated in terms of open surjections. This, however, suffices for establishing the existence of a comparison map between the paths and the coverings fundamental groups in the general case.
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"... Abstract. Let S be an essentially smooth scheme over a field of characteristic exponent c. We prove that there is a canonical equivalence of motivic spectra over S MGL/(a1, a2,...)[1/c] ' HZ[1/c], where HZ is the motivic cohomology spectrum, MGL is the algebraic cobordism spectrum, and the ele ..."
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Abstract. Let S be an essentially smooth scheme over a field of characteristic exponent c. We prove that there is a canonical equivalence of motivic spectra over S MGL/(a1, a2,...)[1/c] ' HZ[1/c], where HZ is the motivic cohomology spectrum, MGL is the algebraic cobordism spectrum, and the elements an are generators of the Lazard ring. We discuss several applications including the computation of the slices of Z[1/c]-local Landweber exact motivic spectra and the convergence of the associated slice spectral sequences.
