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39
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 47 (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
Hypercovers and simplicial presheaves
 MATH. PROC. CAMBRIDGE PHILOS. SOC
, 2004
"... We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly describe ..."
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Cited by 34 (6 self)
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We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition. These ideas are central to constructing realization functors on the homotopy theory of schemes [DI1, Is]. We give a few other applications for this new description of the homotopy theory of simplicial presheaves.
Classical motivic polylogarithm according to Beilinson and Deligne
 DOC. MATH. J. DMV
, 1998
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A closed model structure for ncategories, internal Hom, nstacks and generalized SeifertVan Kampen
, 1997
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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 ...
Cocycle categories
 In Algebraic Topology
, 2009
"... Suppose that G is a sheaf of groups on a space X and that Uα ⊂ X is an open covering. Then a cocycle for the covering is traditionally defined to be a family of elements gαβ ∈ G(Uα ∩ Uβ) such that gαα = e and gαβgβγ = gαγ when all elements are restricted to the group G(Uα ∩ Uβ ∩ Uγ). ..."
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Cited by 9 (5 self)
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Suppose that G is a sheaf of groups on a space X and that Uα ⊂ X is an open covering. Then a cocycle for the covering is traditionally defined to be a family of elements gαβ ∈ G(Uα ∩ Uβ) such that gαα = e and gαβgβγ = gαγ when all elements are restricted to the group G(Uα ∩ Uβ ∩ Uγ).
Algebraic (geometric) nstacks
"... In the introduction of LaumonMoretBailly ([LMB] p. 2) they refer to a possible theory of algebraic nstacks: Signalons au passage que Grothendieck propose d’élargir à son tour le cadre précédent en remplaçant les 1champs par des nchamps (grosso modo, des faisceaux en ncatégories sur (Aff) ou su ..."
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Cited by 8 (2 self)
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In the introduction of LaumonMoretBailly ([LMB] p. 2) they refer to a possible theory of algebraic nstacks: Signalons au passage que Grothendieck propose d’élargir à son tour le cadre précédent en remplaçant les 1champs par des nchamps (grosso modo, des faisceaux en ncatégories sur (Aff) ou sur un site arbitraire) et il ne fait guère de doute qu’il existe une notion utile de nchamps algébriques.... The purpose of this paper is to propose such a theory. I guess that the main reason why Laumon and MoretBailly didn’t want to get into this theory was for fear of getting caught up in a horribly technical discussion of nstacks of groupoids over a general site. In this paper we simply assume that a theory of nstacks of groupoids exists. This is not an unreasonable assumption, first of all because there is a relatively good substitute—the theory of simplicial presheaves or presheaves of spaces ([Bro] [BG] [Jo] [Ja] [Si3] [Si2])— which should be equivalent, in an appropriate sense, to any eventual theory of nstacks; and second of all because it seems likely that a real theory of nstacks of ngroupoids could be developped in the near future ([Br2], [Ta]).
Twisted differential String and Fivebrane structures
, 2009
"... Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the GreenSchwarz mechanism. On the other hand, this mechanism, as well as the FreedWitten an ..."
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Cited by 7 (7 self)
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Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the GreenSchwarz mechanism. On the other hand, this mechanism, as well as the FreedWitten anomaly cancellation, are fundamentally governed by the cohomology classes represented by the relevant nonabelian O(n) and U(n)principal bundles underlying the tangent and the gauge bundle on target space. In this article we unify the picture by describing nonabelian differential cohomology and twisted nonabelian differential cohomology and apply it to these situations. We demonstrate that the FreedWitten mechanism for the Bfield, the GreenSchwarz mechanism for the H3field, as well as its magnetic dual version for the H7field define cocycles in twisted nonabelian differential cohomology that may be addressed, respectively, as twisted Spin(n), twisted String(n) and twisted Fivebrane(n)structures on target space, where the twist in each case is provided by the obstruction to lifting the gauge bundle through a higher connected cover of U(n). We work out the (nonabelian) L∞algebra valued connection data provided by the differential refinements of these twisted cocycles and demonstrate that this reproduces locally the differential form data with the twisted Bianchi identities as known from the
Model structures for homotopy of internal categories
 Theory Appl. Categ
"... CatC of internal categories and functors in a given finitely complete categoryC. Several nonequivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C. Under mild conditions on C, the regular epimorphis ..."
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Cited by 5 (0 self)
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CatC of internal categories and functors in a given finitely complete categoryC. Several nonequivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C. Under mild conditions on C, the regular epimorphism topology determines a modelstructure where we is the class of weak equivalences of internal categories (in the sense of Bunge and Par'e). For a Grothendieck topos C we get a structure that, thoughdifferent from Joyal and Tierney's, has an equivalent homotopy category. In case C is semiabelian, these weak equivalences turn out to be homology isomorphisms, and themodel structure on CatC induces a notion of homotopy of internal crossed modules. Incase C is the category