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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].
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 7 (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
A generalization of Quillen’s small object argument
 J. Pure Appl. Algebra
"... Abstract. We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen’s small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were ..."
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Cited by 7 (3 self)
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Abstract. We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen’s small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were proven to be noncofibrantly generated [2, 6, 8, 20]. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces [10] and diagrams of chain complexes. We also formulate a nonfunctorial version of the argument, which applies in two different model structures on the category of prospaces [11, 20]. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a classcofibrantly
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.
ETALE HOMOTOPY AND SUMSOFSQUARES FORMULAS
"... Abstract. This paper uses a relative of BPcohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sumsofsquares formulas over fields of characteristic p> 2. These conditions were previously known in characteristic zero ..."
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Cited by 3 (0 self)
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Abstract. This paper uses a relative of BPcohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sumsofsquares formulas over fields of characteristic p> 2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized étale cohomology theory called étale BP 2. Contents
Obstruction theory in model categories
 ADV. MATH
, 2004
"... Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Workingin ..."
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Cited by 2 (1 self)
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Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Workingin an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.
SOME REMARKS ON PROFINITE COMPLETION OF SPACES
"... Abstract. We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of ArtinMazur and Sullivan which extends also to nonconnected spaces. Another new aspect is an equivariant profinite completion functor and equivari ..."
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Abstract. We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of ArtinMazur and Sullivan which extends also to nonconnected spaces. Another new aspect is an equivariant profinite completion functor and equivariant fibrant replacement functor for a profinite group acting on a space. This is crucial for applications where, for example, Galois groups are involved, or for profinite Teichmüller theory where equivariant completions are applied. Along the way we collect and survey the most important known results about profinite completion of spaces. 1.
www.elsevier.com/locate/aim Stable étale realization and étale cobordism
, 2006
"... 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 A1homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realizatio ..."
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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 A1homotopy 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 Atiyah–Hirzebruch 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.