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Combinatorial model categories have presentations
 Adv. in Math. 164
, 2001
"... Abstract. We show that every combinatorial model category is Quillen equivalent to a localization of a diagram category (where ‘diagram category’ means diagrams of simplicial sets). This says that every combinatorial model ..."
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Cited by 52 (7 self)
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Abstract. We show that every combinatorial model category is Quillen equivalent to a localization of a diagram category (where ‘diagram category’ means diagrams of simplicial sets). This says that every combinatorial model
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 38 (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
Topological hypercovers and A¹realizations
, 2004
"... We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U∗→X is a weak equivalence. This fact is used to construct topological realization functors for the A1homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about co ..."
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We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U∗→X is a weak equivalence. This fact is used to construct topological realization functors for the A1homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.
HYPERCOVERS IN TOPOLOGY
"... Abstract. We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U ∗ → X is a weak equivalence. This fact is used to construct topological realization functors for the A1homotopy theory of schemes over real and complex fields. 1. ..."
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Abstract. We show that if U ∗ is a hypercover of a topological space X then the natural map hocolim U ∗ → X is a weak equivalence. This fact is used to construct topological realization functors for the A1homotopy theory of schemes over real and complex fields. 1.