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136
Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Cited by 53 (0 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
Universal homotopy theories
 Adv. Math
"... Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy the ..."
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Cited by 37 (3 self)
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Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents
Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 22 (6 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
A model structure on the category of prosimplicial sets
 Trans. Amer. Math. Soc
"... Abstract. We study the category proSS of prosimplicial sets, which arises in étale homotopy theory, shape theory, and profinite completion. We establish a model structure on proSS so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict ..."
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Cited by 20 (6 self)
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Abstract. We study the category proSS of prosimplicial sets, which arises in étale homotopy theory, shape theory, and profinite completion. We establish a model structure on proSS so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for prospaces we use local systems on prospaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for indspaces.
Strict model structures for pro–categories. Categorical decomposition techniques in algebraic topology
 Isle of Skye
, 2004
"... Abstract. We show that if C is a proper model category, then the procategory proC has a strict model structure in which the weak equivalences are the levelwise weak equivalences. This is related to a major result of [10]. The strict model structure is the starting point for many homotopy theories ..."
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Cited by 19 (4 self)
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Abstract. We show that if C is a proper model category, then the procategory proC has a strict model structure in which the weak equivalences are the levelwise weak equivalences. This is related to a major result of [10]. The strict model structure is the starting point for many homotopy theories of proobjects such as those described in [5], [17], and [19].
Coarse Alexander duality and duality groups
 JOURNAL OF DIFFERENTIAL GEOMETRY
, 1999
"... We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplic ..."
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Cited by 18 (5 self)
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We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action G � X determines a collection of “peripheral ” subgroups H1,..., Hk ⊂ G so that the group pair (G, {H1,..., Hk}) is an ndimensional Poincare duality pair. In particular, if G is a 2dimensional 1ended group of type F P2, and G � X is a free simplicial action on a coarse P D(3) space X, then G contains surface subgroups; if in addition X is simply connected, then we obtain a partial generalization of the Scott/Shalen compact core theorem to the setting of coarse P D(3) spaces. In the process we develop coarse topological language and a formulation of coarse Alexander duality which is suitable for applications involving quasiisometries and geometric group theory.
On the Ktheory spectrum of a ring of algebraic integers
 Journal of Ktheory
, 1998
"... Suppose that F is a number field (i.e. a finite algebraic extension of the field Q of rational numbers) and that OF is the ring of algebraic integers in F. One of the most fascinating and apparently difficult problems in algebraic Ktheory is to compute the groups KiOF. These groups were shown to be ..."
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Cited by 16 (5 self)
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Suppose that F is a number field (i.e. a finite algebraic extension of the field Q of rational numbers) and that OF is the ring of algebraic integers in F. One of the most fascinating and apparently difficult problems in algebraic Ktheory is to compute the groups KiOF. These groups were shown to be finitely generated by
Differentiable Stacks and Gerbes
, 2008
"... We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹bundles and S¹gerbes over differentiable stacks. In particular, we establish the relationship between S¹gerbes and groupoid S¹central extensions. We define connections and curvings for groupoid S¹ ..."
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Cited by 14 (3 self)
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We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹bundles and S¹gerbes over differentiable stacks. In particular, we establish the relationship between S¹gerbes and groupoid S¹central extensions. We define connections and curvings for groupoid S¹central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S¹gerbes over manifolds. We develop a ChernWeil theory of characteristic classes in this general setting by presenting a construction of Chern classes and DixmierDouady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S¹bundles and S¹gerbes extending the wellknown result of Weil and Kostant. In particular, we give an explicit construction of S¹central extensions with prescribed curvaturelike data.