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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.
Algebraic Ktheory of topological Ktheory
"... Let ℓp be the pcomplete connective Adams summand of topological Ktheory, with coefficient ring (ℓp) ∗ = Zp[v1], and let V (1) be the Smith–Toda complex, with BP∗(V (1)) = BP∗/(p, v1). For p ≥ 5 we explicitly compute the V (1)homotopy of the algebraic Ktheory spectrum of ℓp, denoted V (1)∗K(ℓp ..."
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Cited by 21 (10 self)
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Let ℓp be the pcomplete connective Adams summand of topological Ktheory, with coefficient ring (ℓp) ∗ = Zp[v1], and let V (1) be the Smith–Toda complex, with BP∗(V (1)) = BP∗/(p, v1). For p ≥ 5 we explicitly compute the V (1)homotopy of the algebraic Ktheory spectrum of ℓp, denoted V (1)∗K(ℓp). In particular we find that it is a free finitely generated module over the polynomial algebra P (v2), except for a sporadic class in degree 2p − 3. Thus also in this case algebraic Ktheory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic Ktheory to topological cyclic homology, and the calculation is
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 15 (9 self)
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Roughly speaking, the stable homotopy category of algebraic topology is obtained from the
PRODUCT AND OTHER FINE STRUCTURE IN POLYNOMIAL RESOLUTIONS OF MAPPING SPACES
, 2001
"... Let MapT (K, X) denote the mapping space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space X to the suspension spectrum Σ ∞ MapT (K, X). Applying a g ..."
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Cited by 11 (6 self)
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Let MapT (K, X) denote the mapping space of continuous based functions between two based spaces K and X. If K is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space X to the suspension spectrum Σ ∞ MapT (K, X). Applying a generalized homology theory h ∗ to this tower yields a spectral sequence, and this will converge strongly to h∗(MapT (K, X)) under suitable conditions, e.g. if h ∗ is connective and X is at least dim K connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable light on h∗(MapT (K, X)). Similar comments hold when a cohomology theory is applied. In this paper we study how various important natural constructions on mapping spaces induce extra structure on the towers. This leads to useful interesting additional structure in the associated spectral sequences. For example, the diagonal on MapT (K, X) induces a ‘diagonal ’ on the associated tower. After applying any cohomology theory with products h ∗ , the resulting spectral sequence is then a spectral sequence of differential graded algebras. The product on the E∞–term corresponds to the cup product in h ∗ (MapT (K, X)) in the usual way, and the product on the E1–term is described in terms of group theoretic transfers. We use explicit equivariant S–duality maps to show that, when K is the sphere S n, our constructions at the fiber level have descriptions in terms of the Boardman–Vogt little n–cubes spaces. We are then able to identify, in a computationally useful way, the Goodwillie tower of the functor from spectra to spectra sending a spectrum X to Σ ∞ Ω ∞ X.
Iterated wreath product of the simplex category and iterated loop spaces
 Adv. Math
"... Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternat ..."
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Cited by 11 (4 self)
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Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γspace. In particular, each EilenbergMacLane space has a canonical reduced Θnset model. The number of (n + d)dimensional cells of the resulting CWcomplex of type K(Z/2Z, n) is a generalised Fibonacci number.
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.
THE MCCORD MODEL FOR THE TENSOR PRODUCT OF A SPACE AND A COMMUTATIVE RING SPECTRUM
, 2002
"... We begin this paper by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal’s very special Γ–spaces indeed this is ..."
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Cited by 6 (3 self)
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We begin this paper by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal’s very special Γ–spaces indeed this is roughly what Segal did and then to a more modern situation: K ⊗ R where K is a based space and R is a unital, augmented, commutative, associative S–algebra. The model comes with an easytodescribe filtration. If one lets K = S n, and then stabilize with respect to n, one gets a filtered model for the Topological André–Quillen Homology of R. When R = Ω ∞ Σ ∞ X, one arrives at a filtered model for the connective cover of a spectrum X, constructed from its 0 th space. Another example comes by letting K be a finite complex, and R the S–dual of a finite complex Z. Dualizing again, one arrives at G. Arone’s model for the Goodwillie tower of the functor sending Z to Σ ∞ MapT (K, Z). Applying cohomology with field coefficients, one gets various spectral sequences for deloopings with known E1–terms. A few nontrivial examples are given. In an appendix, we describe the construction for unital, commutative, associative S–algebras not necessarily augmented.
On the homotopy groups of symmetric spectra
 Geom. Topol
"... Symmetric spectra are an easytodefine and convenient model for the stable homotopy category with a nice smash product. Symmetric ring spectra first showed up under the name ‘FSP on spheres ‘ in the context of algebraic Ktheory and topological Hochschild homology. Around 1993, Jeff Smith made the ..."
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Cited by 5 (0 self)
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Symmetric spectra are an easytodefine and convenient model for the stable homotopy category with a nice smash product. Symmetric ring spectra first showed up under the name ‘FSP on spheres ‘ in the context of algebraic Ktheory and topological Hochschild homology. Around 1993, Jeff Smith made the crucial observation that the ‘FSP on spheres ‘ are the monoids in a category of ‘symmetric spectra ‘ with respect to an associative and commutative smash prodct, and he suspected compatible model category structures so that one obtains as homotopy categories ‘the ‘ stable homotopy category (for symmetric spectra), the homotopy category of A ∞ ring spectra (for symmetric ring spectra), respectively the homotopy category of E ∞ ring spectra (for commutative symmetric ring spectra). The details of various model structures were worked out by Hovey, Shipley and Smith in [HSS]. Maybe the only tricky point with symmetric spectra is that the stable equivalences can not be defined by looking at stable homotopy groups (defined as the classical sequential colimit of the unstable homotopy groups of the terms in a symmetric spectrum). Formally inverting the π∗isomorphisms, i.e., those morphisms which induce isomorphisms
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.