Results 1  10
of
19
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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Cited by 14 (0 self)
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Simplicial cohomology of orbifolds
 Indag. Math. (N.S
, 1999
"... For any orbifold M, we explicitly construct a simplicial complex S(M) from a given triangulation of the ‘coarse ’ underlying space together with the local isotropy groups of M. We prove that, for any local system on M, this complex S(M) has the same cohomology as M. The use of S(M) in explicit calcu ..."
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Cited by 10 (0 self)
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For any orbifold M, we explicitly construct a simplicial complex S(M) from a given triangulation of the ‘coarse ’ underlying space together with the local isotropy groups of M. We prove that, for any local system on M, this complex S(M) has the same cohomology as M. The use of S(M) in explicit calculations is illustrated in the example of the ‘teardrop ’ orbifold. Introduction. Orbifolds or Vmanifolds were first introduced by Satake [9], and arise naturally in many ways. For example, the orbit space of any proper action by a (discrete) group on a manifold has the structure of an orbifold; this applies in particular to moduli spaces. Furthermore, the orbit space of any almost free action by a
Topological and smooth stacks
"... Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are ..."
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Cited by 8 (0 self)
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Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces). 1.
NonHausdorff Groupoids, Proper Actions and KTheory
 DOCUMENTA MATH.
, 2004
"... Let G be a (not necessarily Hausdorff) locally compact groupoid. We introduce a notion of properness for G, which is invariant under Moritaequivalence. We show that any generalized morphism between two locally compact groupoids which satisfies some properness conditions induces a C∗correspondence ..."
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Cited by 8 (0 self)
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Let G be a (not necessarily Hausdorff) locally compact groupoid. We introduce a notion of properness for G, which is invariant under Moritaequivalence. We show that any generalized morphism between two locally compact groupoids which satisfies some properness conditions induces a C∗correspondence from C ∗ r(G2) to C ∗ r(G1), and thus two Morita equivalent groupoids have Moritaequivalent C∗algebras.
Operator Algebras and Poisson Manifolds Associated to Groupoids
, 2001
"... It is well known that a measured groupoid G defines a von Neumann algebra W # (G), and that a Lie groupoid G canonically defines both a C # algebra C # (G) and a Poisson manifold A # (G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C # algebras, and ..."
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Cited by 7 (3 self)
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It is well known that a measured groupoid G defines a von Neumann algebra W # (G), and that a Lie groupoid G canonically defines both a C # algebra C # (G) and a Poisson manifold A # (G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C # algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G C # (G), and G are functorial between the categories in question. It follows that these maps preserve Morita equivalence.
Homology of formal deformations of proper étale Lie groupoids
, 2005
"... In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multivector fields on the associat ..."
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Cited by 6 (5 self)
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In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multivector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all
Quantization as a functor
 in ”Quantization, Poisson Brackets and beyond”, Contemp. Math.,315, AMS
, 2002
"... “First quantization is a mystery, but second quantization is a functor” ..."
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Cited by 4 (1 self)
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“First quantization is a mystery, but second quantization is a functor”
Crystallographic Topology 2: Overview And Work In Progress
, 1999
"... This overview describes an application of contemporary geometric topology and stochastic process concepts to structural crystallography. In this application, crystallographic groups become orbifolds, crystal structures become Morse functions on orbifolds, and vibrating atoms in a crystal become vect ..."
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This overview describes an application of contemporary geometric topology and stochastic process concepts to structural crystallography. In this application, crystallographic groups become orbifolds, crystal structures become Morse functions on orbifolds, and vibrating atoms in a crystal become vector valued Gaussian measures with the RadonNikodym property. Intended crystallographic benefits include new methods for visualization of space groups and crystal structures, analysis of the thermal motion patterns seen in ORTEP drawings, and a classification scheme for crystal structures based on their Heegaard splitting properties. 1 Introduction Geometric topology and structural crystallography concepts are combined to define a research area we call Structural Crystallographic Topology, or just Crystallographic Topology. The first paper in the series[30] describes basic crystallography concepts (crystallographic groups, lattice complexes, and crystal structures) and their replacement topo...