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Duality and ProSpectra
, 2004
"... Abstract Cofiltered diagrams of spectra, also called prospectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of prospectra and to study its relation to the usual homotopy theory of spectra ..."
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Cited by 6 (1 self)
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Abstract Cofiltered diagrams of spectra, also called prospectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of prospectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of prospectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of SpanierWhitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the SpanierWhitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the SpanierWhitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of indspectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories.
Stable étale realization and étale cobordism
 Adv. Math
"... We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A 1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realizati ..."
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Cited by 4 (4 self)
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We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A 1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero. On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an AtiyahHirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element. 1
Profinite Homotopy Theory
 DOCUMENTA MATH.
, 2008
"... We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and prospaces. One motivation is the étale homotopy theory of schemes in which higher profinite étale homotopy g ..."
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Cited by 3 (3 self)
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We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and prospaces. One motivation is the étale homotopy theory of schemes in which higher profinite étale homotopy groups fit well with the étale fundamental group which is always profinite. We show that the profinite étale topological realization functor is a good object in several respects.
DOI: 10.1017/S0305004103007175 Printed in the United Kingdom Hypercovers and simplicial presheaves
, 2002
"... 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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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, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves. 1.
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.