Results 1  10
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17
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].
Homotopy limits and colimits and enriched homotopy theory
, 2006
"... Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equiv ..."
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Cited by 15 (2 self)
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Abstract. Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, showing that the classical explicit constructions still give the right answer in the abstract sense, thus partially bridging the gap between classical homotopy theory and modern abstract homotopy theory. To do this we introduce a notion of “enriched homotopical categories”, which are more general than enriched model categories, but are still a good place to do enriched homotopy theory. This demonstrates that the presence of enrichment often simplifies rather than complicates matters, and goes some way toward achieving a better understanding of “the role of homotopy in homotopy theory.” Contents
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.
EQUIVARIANT HOMOTOPY THEORY FOR ProSpectra
, 2006
"... We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G−homotopy theory is “pieced together” from the G/U−homotopy theories for suitable quotient groups G/U of G; a motivation is th ..."
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Cited by 5 (1 self)
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We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The G−homotopy theory is “pieced together” from the G/U−homotopy theories for suitable quotient groups G/U of G; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In this category Postnikov towers are studied from a general perspective. We introduce pro−G−spectra and construct various model structures on them. A key property of the model structures is that prospectra are weakly equivalent to their Postnikov towers. We give a careful discussion of two versions of a model structure with “underlying weak equivalences”. One of the versions only makes sense for pro−spectra. In the end we use the theory to study homotopy fixed points of pro−Gspectra.
Localization with respect to a class of maps I – Equivariant localization of diagrams of spaces
"... Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3, 12, 20, 29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model cat ..."
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Cited by 4 (4 self)
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Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3, 12, 20, 29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces [11]. This model category is not cofibrantly generated [7]. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the class of maps which induces ordinary localizations on the generalized fixedpoints sets.
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
Localization with respect to a class of maps. II. Equivariant cellularization and its application
 Israel J. Math
"... Abstract. We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result. ..."
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Cited by 3 (3 self)
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Abstract. We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result.
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.