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17
Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... 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 76 (16 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.
Universal homotopy theories
 Adv. Math
"... Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy the ..."
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Cited by 37 (3 self)
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Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the DwyerKan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents
HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Cited by 32 (10 self)
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
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 7 (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.
CALCULUS OF FUNCTORS, OPERAD FORMALITY, AND RATIONAL HOMOLOGY OF EMBEDDING SPACES
, 2007
"... Abstract. Let M be a smooth manifold and V a Euclidean space. Let Emb(M, V) be the homotopy fiber of the map Emb(M, V) − → Imm(M, V). This paper is about the rational homology of Emb(M, V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M, V) ↦ → HQ ∧ Emb(M, V) ..."
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Cited by 4 (3 self)
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Abstract. Let M be a smooth manifold and V a Euclidean space. Let Emb(M, V) be the homotopy fiber of the map Emb(M, V) − → Imm(M, V). This paper is about the rational homology of Emb(M, V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M, V) ↦ → HQ ∧ Emb(M, V)+. Our main theorem states that if dim V ≥ 2ED(M) + 1 (where ED(M) is the embedding dimension of M), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E 1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich’s theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor HQ ∧ Emb(M, V)+. The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of Emb(−, V).
The product formula in unitary deformation Ktheory
 KTheory
"... Abstract. For finitely generated groups G and H, we prove that there is a weak equivalence KG ∧ku KH ≃ K(G × H) of kualgebra spectra, where K denotes the “unitary deformation Ktheory ” functor. Additionally, we give spectral sequences for computing the homotopy groups of KG and HZ ∧ku KG in terms ..."
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Cited by 4 (1 self)
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Abstract. For finitely generated groups G and H, we prove that there is a weak equivalence KG ∧ku KH ≃ K(G × H) of kualgebra spectra, where K denotes the “unitary deformation Ktheory ” functor. Additionally, we give spectral sequences for computing the homotopy groups of KG and HZ ∧ku KG in terms of connective Ktheory and homology of spaces of Grepresentations. 1.
Completed representation ring spectra of nilpotent groups
 ALGEBRAIC & GEOMETRIC TOPOLOGY
"... In this paper, we examine the “derived completion” of the representation ring of a prop group G ∧ p with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the EilenbergMacLane spectrum HZ, and can have higher homoto ..."
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Cited by 4 (2 self)
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In this paper, we examine the “derived completion” of the representation ring of a prop group G ∧ p with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the EilenbergMacLane spectrum HZ, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor R[−] from groups to ring spectra, and show that the map R[G ∧ p] → R[G] becomes an equivalence after completion when G is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the padic Heisenberg group.
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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Cited by 4 (1 self)
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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
Generalized AndréQuillen cohomology
 J. Homotopy Relat. Struct
"... Abstract. We explain how the approach of André and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them. ..."
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Cited by 1 (0 self)
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Abstract. We explain how the approach of André and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them.