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A uniqueness theorem for stable homotopy theory
 Math. Z
, 2002
"... Roughly speaking, the stable homotopy category of algebraic topology is obtained from the ..."
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Cited by 14 (9 self)
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Roughly speaking, the stable homotopy category of algebraic topology is obtained from the
Stable Homotopy of Algebraic Theories
 Topology
, 2001
"... The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic t ..."
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Cited by 12 (1 self)
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The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co)homology of an algebraic theory is isomorphic to the topological Hochschild (co)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...
SemiTopological KTheory of Real VARIETIES
"... . The semitopological Ktheory of real varieties, KR semi (), is an oriented multiplicative (generalized) cohomology theory which extends the authors' earlier theory, K semi (), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles ( ..."
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Cited by 8 (3 self)
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. The semitopological Ktheory of real varieties, KR semi (), is an oriented multiplicative (generalized) cohomology theory which extends the authors' earlier theory, K semi (), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpened to real semitopological equivalence), consideration of Z/2equivariant mapping spaces of morphisms of algebraic varieties to Grassmannian varieties, and consideration of the algebraic Ktheory of real varieties. The authors verify that the semitopological Ktheory of a real variety X interpolates between the algebraic Ktheory of X and Atiyah's Real Ktheory of the associated Real space of complex points, X R (C). The resulting natural maps of spectra K alg (X) # KR semi (X) # KRtop (X R (C)) satisfy numerous good properties: the first map is a modn equivalence for any projective real variety and any n > 0; the second map is an equivalence for smooth projective curves...
Classification of Stable Model Categories
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 6 (5 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R. 1.
Techniques, computations, and conjectures for semitopological Ktheory
 MATH. ANN
, 2004
"... We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence tha ..."
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Cited by 6 (1 self)
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We establish the existence of an “AtiyahHirzebruchlike” spectral sequence relating the morphic cohomology groups of a smooth, quasiprojective complex variety to its semitopological Kgroups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that connects the motivic cohomology and algebraic Ktheory of varieties, and it is also compatible with the classical AtiyahHirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the BorelMoore (singular) cohomology of complex varieties introduced by H. Gillet and C. Soulé — to compute the semitopological Ktheory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational threefolds, and related varieties, the semitopological Kgroups and topological Kgroups are isomorphic in all degrees permitted by cohomological considerations. We also
Rational Isomorphisms between KTheories and Cohomology Theories
"... The well known isomorphism relating the rational algebraic Ktheory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the "Segre map") of infinite loop spaces. Moreover, the associated Chern character map on ratio ..."
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The well known isomorphism relating the rational algebraic Ktheory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the "Segre map") of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced which establishes a useful general criterion for a natural transformation of functors on quasiprojective complex varieties to induce a homotopy equivalence of semitopological singular complexes. Since semitopological Ktheory and morphic cohomology can be formulated as the semitopological singular complexes associated to Ktheory and motivic cohomology, this criterion provides a rational isomorphism between the semitopological Ktheory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a RiemannRoch theorem for the Chern character on semitopological Ktheory and an interpretation of the "topological filtration" on singular cohomology groups in K theoretic terms.
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.