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38
A telescope comparison lemma for THH
, 2002
"... Abstract. We extend to the nonconnective case a lemma of Bökstedt about the equivalence of the telescope with a more complicated homotopy colimit of symmetric spectra used in the construction of THH. 1. ..."
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Abstract. We extend to the nonconnective case a lemma of Bökstedt about the equivalence of the telescope with a more complicated homotopy colimit of symmetric spectra used in the construction of THH. 1.
Fixed point theory and trace for bicategories
, 2007
"... The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point inde ..."
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The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. We define a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. We show the invariants used in the converse of the Lefschetz fixed point theorem are examples of this trace and that the functoriality of the trace provides some of the necessary identifications. The methods used here do not use simplicial techniques and so generalize readily to other contexts. iii Contents
The Classifying Space of a Topological 2Group
, 2008
"... Categorifying the concept of topological group, one obtains the notion of a ‘topological 2group’. This in turn allows a theory of ‘principal 2bundles’ generalizing the usual theory of principal bundles. It is wellknown that under mild conditions on a topological group G and a space M, principal G ..."
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Categorifying the concept of topological group, one obtains the notion of a ‘topological 2group’. This in turn allows a theory of ‘principal 2bundles’ generalizing the usual theory of principal bundles. It is wellknown that under mild conditions on a topological group G and a space M, principal Gbundles over M are classified by either the Čech cohomology ˇ H 1 (M, G) or the set of homotopy classes [M, BG], where BG is the classifying space of G. Here we review work by Bartels, Jurčo, Baas–Bökstedt–Kro, and others generalizing this result to topological 2groups and even topological 2categories. We explain various viewpoints on topological 2groups and the Čech cohomology ˇ H 1 (M, G) with coefficients in a topological 2group G, also known as ‘nonabelian cohomology’. Then we give an elementary proof that under mild conditions on M and G there is a bijection ˇH 1 (M, G) ∼ = [M, BG] where BG  is the classifying space of the geometric realization of the nerve of G. Applying this result to the ‘string 2group ’ String(G) of a simplyconnected compact simple Lie group G, it follows that principal String(G)2bundles have rational characteristic classes coming from elements of H ∗ (BG, Q)/〈c〉, where c is any generator of H 4 (BG, Q).
Effective generalized SeifertVan Kampen: how to calculate ΩX, preprint available at qalg/9710011
"... A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including highe ..."
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A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including higher order coherences as first investigated in [33]. A delooping machine is a specification of the extra homotopical structure carried by the loop space ΩX of a connected basepointed topological space X, exactly the structure allowing recovery of X by a “classifying space ” construction. The first level of structure is that the component set π0(ΩX) has a structure of group π1(X, x). Classically the SeifertVan Kampen theorem states that a pushout diagram of connected spaces gives rise to a pushout diagram of groups π1. The loop space construction ΩX with its delooping structure being the higherorder “topologized ” generalization of π1, an obvious question is whether a similar SeifertVan Kampen statement holds for ΩX. The aim of this paper is to describe the operation underlying pushout of spaces with loop space structure, answering the above question by giving a SeifertVan Kampen statement for delooping machinery. We work with Segal’s machine [28] [36]. Our SeifertVan
Parametrized spaces model locally constant homotopy sheaves
 Topology Appl
, 2008
"... Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification ..."
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Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopytheoretic version of the correspondence between covering spaces and π1sets. We then use these two equivalences to study base change functors for parametrized spaces. Contents
Units of ring spectra and Thom spectra
"... Abstract. We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E ∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor To a map of spectra Σ ∞ + Ω ∞ : ho(c ..."
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Abstract. We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E ∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor To a map of spectra Σ ∞ + Ω ∞ : ho(connective spectra) → ho(E ∞ ring spectra). f: b → bgl1A, we associate an E ∞ Aalgebra Thom spectrum Mf, which admits an E ∞ Aalgebra map to R if and only if the composition b → bgl1A → bgl1R is null; the classical case developed by [MQRT77] arises when A is the sphere spectrum. We develop the analogous theory for A ∞ ring spectra. If A is an A ∞ ring spectrum, then to a map of spaces f: B → BGL1A we associate an Amodule Thom spectrum Mf, which admits an Rorientation if and only if
Equivariant algebraic Ktheory
"... There are many ways that group actions enter into algebraic Ktheory and there are various theories that fit under the rubric of our title. To anyone familiar with both equivariant topological Ktheory and Quillen's original definition and calculations of algebraic Ktheory, there is a perfectly obv ..."
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There are many ways that group actions enter into algebraic Ktheory and there are various theories that fit under the rubric of our title. To anyone familiar with both equivariant topological Ktheory and Quillen's original definition and calculations of algebraic Ktheory, there is a perfectly obvious program for the definition of the equivariant algebraic Ktheory of rings and its calculation for finite fields. While this program surely must have occurred to others, there are no published accounts and the technical details have not been worked out before. That part of the program which pertains to the complex Adams conjecture was outlined in a letter to one of us from Graeme Segal, and the real analog was assumed without proof by tom Dieck [10,11.3.8]. Negatively indexed equivariant Kgroups were introduced by Loday [19]. From a topological point of view, one way of thinking about Quillen's original definition runs as follows. Let ~ be a topological group, perhaps discrete. One has a notion of a principal Kbundle and a classifying space B ~ for such bundles. When H is discrete, a principal ~bundle is just a covering (possibly with disconnected total space) with fibre and group ~. Given any increasing sequence of groups N with union H, we obtain an increasing sequence of classifying spaces n BH with union B~. We then think of B ~ as a classifying space for stable
On the construction of functorial factorizations for model categories
, 2012
"... Abstract. We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic ” characterizations of fibrations to produce factorizations that have the desired lifting properties in a com ..."
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Abstract. We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic ” characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored, and cotensored in spaces, proving the existence of Hurewicztype model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) Gspaces, and diagram spectra among others. 1.
Stratified fibre bundles
, 2003
"... A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the ..."
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A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem. AMS SC: 55R55 (Fiberings with singularities); 55R65 (Generalizations of fiber spaces and bundles); 55R70 (Fiberwise topology); 55R10 (Fibre bundles); 18F15 (Abstract manifolds and fibre bundles); 54H15 (Transformation groups and semigroups); 57S05 (Topological properties of groups of homeomorphisms or diffeomorphisms).