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On the representation theory of Galois and atomic topoi
 J. Pure Appl. Algebra
"... introduction ..."
FirstOrder Logical Duality
, 2008
"... Generalizing Stone duality for Boolean algebras, an adjunction between Boolean coherent categories—representing firstorder syntax—and certain topological groupoids—representing semantics—is constructed. The embedding of a Boolean algebra into a frame of open sets of a space of 2valued models is re ..."
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Generalizing Stone duality for Boolean algebras, an adjunction between Boolean coherent categories—representing firstorder syntax—and certain topological groupoids—representing semantics—is constructed. The embedding of a Boolean algebra into a frame of open sets of a space of 2valued models is replaced by an embedding of a Boolean coherent category, B, into a topos of equivariant sheaves on a topological groupoid of setvalued models and isomorphisms between them. The latter is a groupoid representation of the topos of coherent sheaves on B, analogously to how the Stone space of a Boolean algebra is a spatial representation of the ideal completion of the algebra, and the category B can then be recovered from its semantical groupoid, up to pretopos completion. By equipping the groupoid of sets and bijections with a particular topology, one obtains a particular topological groupoid which plays a role analogous to that of the discrete space 2, in being the dual of the object classifier and the object one ‘homs into ’ to recover a Boolean coherent category from its semantical groupoid. Both parts of the adjunction, then, consist of ‘homming into sets’, similarly to how both parts of the equivalence between Boolean algebras and Stone spaces consist of ‘homming into 2’. By slicing over this groupoid (modified to display an alternative setup), Chapter 3 shows how the adjunction specializes to the case of firstorder single sorted logic to yield an adjunction between such theories and an independently characterized slice category of topological groupoids such that the counit component at a theory is an isomorphism. Acknowledgements I would like, first and foremost, to thank my supervisor Steve Awodey. I would like to thank the members of the committee: Jeremy Avigad, Lars
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).
The LusternikSchnirelmann category of a Lie groupoid, to appear
 in Transactions of the American Mathematical Society (2009) THE 1HOMOTOPY TYPE OF LIE GROUPOIDS 29
"... Abstract. We propose a new homotopy invariant for Lie groupoids which generalizes the classical LusternikSchnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given b ..."
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Abstract. We propose a new homotopy invariant for Lie groupoids which generalizes the classical LusternikSchnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well defined LScategory for orbifolds. We prove an orbifold version of the classical LusternikSchnirelmann theorem for critical points. 1.
TRANSLATION GROUPOIDS AND ORBIFOLD BREDON COHOMOLOGY
, 705
"... Abstract. We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an example, w ..."
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Abstract. We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an example, we use this result to define an orbifold version of Bredon cohomology. 1.
LOCALIZATIONS FOR CONSTRUCTION OF QUANTUM COSET SPACES
, 2003
"... Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on n ..."
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Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localizations on the categories of Hopf modules. Their interplay with the functor of taking coinvariants and its left adjoint is stressed out. Using localization approach, we constructed a natural class of examples of quantum coset spaces, related to the quantum flag varieties of type A of other authors. Noncommutative Gauss decomposition via quasideterminants reveals a new structure in noncommutative matrix bialgebras. In the quantum case, calculations with quantum minors yield the structure theorems. Notation. Ground field is k and we assume it is of characteristic zero. If we deal just with one kHopf algebra, say B, the comultiplication is ∆ : B → B ⊗ B, unit map η: k → B, counit ǫ: B → k, multiplication µ: B ⊗ B → B, and antipode (coinverse) is
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&quo ..."
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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
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
Documenta Math. 313 On the WeilÉtale Topos of Regular Arithmetic Schemes
, 2010
"... Abstract. We define and study a Weilétale topos for any regular, proper scheme X over Spec(Z) which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with ˜ Rcoefficients has the expected relation to ζ(X,s) at s = 0 if the HasseWeil Lfunctions L( ..."
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Abstract. We define and study a Weilétale topos for any regular, proper scheme X over Spec(Z) which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with ˜ Rcoefficients has the expected relation to ζ(X,s) at s = 0 if the HasseWeil Lfunctions L(h i (XQ),s) have the expected meromorphic continuation and functional equation. If X has characteristic p the cohomology with Zcoefficients also has the expected relation to ζ(X,s) and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.