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44
Postnikov extensions for ring spectra
, 2006
"... We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum. ..."
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We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum.
ENRICHED MODEL CATEGORIES AND AN APPLICATION TO ADDITIVE ENDOMORPHISM SPECTRA
"... Abstract. We define the notion of an additive model category and prove that ..."
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Abstract. We define the notion of an additive model category and prove that
Classification spaces of maps in model categories
"... Abstract. We correct a mistake in [DK2] and use this to identify homotopy function complexes in a model category with the nerves of certain categories of zigzags. ..."
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Cited by 5 (3 self)
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Abstract. We correct a mistake in [DK2] and use this to identify homotopy function complexes in a model category with the nerves of certain categories of zigzags.
Triangulated categories of mixed motives
"... Abstract. We construct triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky’s definition of motives over a field. We prove that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by st ..."
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Abstract. We construct triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky’s definition of motives over a field. We prove that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following insights and constructions of Beilinson, Morel and Voevodsky. Finally, we associate with any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors.
A short note on models for equivariant homotopy theory
, 2006
"... These notes explore equivariant homotopy theory from the perspective of model categories in the case of a discrete group G. Section 2 reviews the situation for topological spaces, largely following [May]. In section 3, we discuss two approaches to equivariant homotopy theory in more general model ca ..."
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These notes explore equivariant homotopy theory from the perspective of model categories in the case of a discrete group G. Section 2 reviews the situation for topological spaces, largely following [May]. In section 3, we discuss two approaches to equivariant homotopy theory in more general model categories. Section 4 discusses
NONCOMMUTATIVE CORRESPONDENCE CATEGORIES, SIMPLICIAL SETS AND PRO C ∗ALGEBRAS
, 906
"... Abstract. We construct an additive functor from the category of separable C ∗algebras with morphisms enriched over Kasparov’s KK0groups to the noncommutative correspondence category NCC K dg, whose objects are small DG categories and morphisms are given by the equivalence classes of some DG bimodu ..."
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Abstract. We construct an additive functor from the category of separable C ∗algebras with morphisms enriched over Kasparov’s KK0groups to the noncommutative correspondence category NCC K dg, whose objects are small DG categories and morphisms are given by the equivalence classes of some DG bimodules up to a certain Ktheoretic identification. Motivated by a construction of Cuntz we associate a pro C ∗algebra to any simplicial set, which is functorial with respect to proper maps of simplicial sets and those of pro C ∗algebras. This construction respects homotopy between proper maps after enforcing matrix stability on the category of pro C ∗algebras. The first result can be used to deduce derived Morita equivalence between DG categories of topological bundles associated to separable C ∗algebras up to a Ktheoretic identification from the knowledge of KKequivalence between the C ∗algebras. The second construction gives an indication that one can possibly develop a noncommutative proper homotopy theory in the context of topological algebras, e.g., pro C ∗algebras.
ON LEFT AND RIGHT MODEL CATEGORIES AND LEFT AND RIGHT BOUSFIELD LOCALIZATIONS
"... We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of ..."
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We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopycoherent descent condition. We then verify the existence of right Bousfield localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.