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37
Categorical homotopy theory
 Homology, Homotopy Appl
"... This paper is an exposition of the ideas and methods of Cisinksi, in the context of Apresheaves on a small ..."
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Cited by 183 (8 self)
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This paper is an exposition of the ideas and methods of Cisinksi, in the context of Apresheaves on a small
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 33 (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.
Cyclic homology, cdhcohomology and negative Ktheory
, 2005
"... We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheor ..."
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Cited by 22 (7 self)
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We prove a blowup formula for cyclic homology which we use to show that infinitesimal Ktheory satisfies cdhdescent. Combining that result with some computations of the cdhcohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic Ktheory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.
Patching the norm residue isomorphism theorem
, 2007
"... Abstract. We provide a patch to complete the proof of the VoevodskyRost Theorem, that the norm residue map is an isomorphism. (This settles the motivic BlochKato conjecture). ..."
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Cited by 17 (2 self)
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Abstract. We provide a patch to complete the proof of the VoevodskyRost Theorem, that the norm residue map is an isomorphism. (This settles the motivic BlochKato conjecture).
Motivic cell structures
 ALGEBR. GEOM. TOPOL
, 2005
"... An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic Ktheory and algebraic cobordism spectra are both cellular, and prove some Kü ..."
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Cited by 11 (1 self)
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An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic Ktheory and algebraic cobordism spectra are both cellular, and prove some Künneth theorems for cellular objects.
Motivic Functors
 DOCUMENTA MATH.
, 2003
"... The notion of motivic functors refers to a motivic homotopy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivale ..."
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Cited by 10 (7 self)
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The notion of motivic functors refers to a motivic homotopy theoretic analog of continuous functors. In this paper we lay the foundations for a homotopical study of these functors. Of particular interest is a model structure suitable for studying motivic functors which preserve motivic weak equivalences and a model structure suitable for motivic stable homotopy theory. The latter model is Quillen equivalent to the category of motivic symmetric spectra. There is a symmetric monoidal smash product of motivic functors, and all model structures constructed are compatible with the smash product in the sense that we can do homotopical algebra on the various categories of modules and algebras. In particular, motivic cohomology
DESCENT PROPERTIES OF HOMOTOPY KTHEORY
"... In this paper, we show that the widely held expectation that Weibel’s homotopy Ktheory satisfies cdhdescent is indeed fulfilled for schemes over a field of characteristic zero. The main ingredient in the proof is a certain factorization of the resolution of hypersurface singularities. Some consequ ..."
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Cited by 9 (4 self)
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In this paper, we show that the widely held expectation that Weibel’s homotopy Ktheory satisfies cdhdescent is indeed fulfilled for schemes over a field of characteristic zero. The main ingredient in the proof is a certain factorization of the resolution of hypersurface singularities. Some consequences are derived. Finally, some evidence for a conjecture of Weibel concerning negative Ktheory is given. 1.
Etale realization on the A¹homotopy theory of schemes
 MATH
, 2001
"... We compare Friedlander’s definition of étale homotopy for simplicial schemes to another definition involving homotopy colimits of prosimplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the ca ..."
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Cited by 6 (3 self)
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We compare Friedlander’s definition of étale homotopy for simplicial schemes to another definition involving homotopy colimits of prosimplicial sets. This can be expressed as a notion of hypercover descent for étale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the étale site of schemes over S to the category of prospaces. After completing away from the characteristics of the
On the homotopy theory of ntypes
 Homology, Homotopy Appl
"... Abstract. An ntruncated model structure on simplicial (pre)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine’s intermediate model structures we construct such an ntype model structure via Bo ..."
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Cited by 3 (1 self)
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Abstract. An ntruncated model structure on simplicial (pre)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine’s intermediate model structures we construct such an ntype model structure via BousfieldFriedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called nhyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analogous ntruncations on simplicial groupoids and prove a Quillen equivalence between these settings. We achieve a classification of ntypes of simplicial presheaves in terms of (n −1)types of presheaves of simplicial groupoids. Our classification holds for general n. Therefore this can also be viewed as the homotopy theory of (pre)sheaves of (weak) higher groupoids. Contents