Results 1  10
of
12
Stabilization of model categories
, 1998
"... monoidal structure which is compatible with the model structure. Given a monoidal model category, we consider the homotopy theory of modules over a given monoid and the homotopy theory of monoids. We make minimal assumptions on our model categories; our results therefore are more general, yet weaker ..."
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Cited by 204 (8 self)
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monoidal structure which is compatible with the model structure. Given a monoidal model category, we consider the homotopy theory of modules over a given monoid and the homotopy theory of monoids. We make minimal assumptions on our model categories; our results therefore are more general, yet weaker, than the results of [10]. In particular, our results apply to the monoidal model category of topological symmetric spectra [7].
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.
Spectra and symmetric spectra in general model categories
 J. Pure Appl. Algebra
"... Abstract. We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A1homotopy theory of MorelVoevodsky [16, 23]. One is based on the standard notion of spectra originated by Boa ..."
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Cited by 55 (0 self)
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Abstract. We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A1homotopy theory of MorelVoevodsky [16, 23]. One is based on the standard notion of spectra originated by Boardman [24]. Its input is a wellbehaved model category C and an endofunctor
A convenient model category for commutative ring spectra
, 2003
"... We develop a new system of model structures on the modules, algebras and commutative algebras over symmetric spectra. In addition to the same properties as the standard stable model structures defined in [HSS] and [MMSS], these model structures have better compatibility properties between commutati ..."
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Cited by 41 (2 self)
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We develop a new system of model structures on the modules, algebras and commutative algebras over symmetric spectra. In addition to the same properties as the standard stable model structures defined in [HSS] and [MMSS], these model structures have better compatibility properties between commutative algebras and the underlying modules.
Simplicial Structures on Model Categories and Functors
 Amer.J.Math.123
, 2001
"... We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model cate ..."
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Cited by 17 (3 self)
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We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.
Monoidal uniqueness of stable homotopy theory
 Adv. in Math. 160
, 2001
"... Abstract. We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlyi ..."
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Cited by 12 (7 self)
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Abstract. We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlying model structures Margolis ’ axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (Smodules, Wspaces, orthogonal spectra, simplicial functors) with symmetric spectra. The equivalences of modules, algebras and commutative algebras in these categories are also considered. 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.
by Paul G. Goerss
, 1996
"... Let A be a simplicial bicommutative Hopf algebra over the field F 2 with the property that 0 A = F 2 . We show that A is a functor of the Andr'eQuillen homology of A, where A is regarded as an F 2 algebra. Then we give a method for calculating that Andr'eQuillen homology independent of know ..."
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Let A be a simplicial bicommutative Hopf algebra over the field F 2 with the property that 0 A = F 2 . We show that A is a functor of the Andr'eQuillen homology of A, where A is regarded as an F 2 algebra. Then we give a method for calculating that Andr'eQuillen homology independent of knowledge of A. Let G be an abelian group. Since the work of Serre [19] and Cartan [6], we have know that the mod p homology of an EilenbergMacLane space K(G;n), n 1, depends only on Tor s (Z=p; G), s = 0; 1. More is true: the structure of H K(G;n) = H (K(G; n); F p ) as an unstable coalgebra over the Steenrod algebra depends only on there Tor groups and the Bockstein fi : Tor 1 (Z=p; G) ! Tor 0 (Z=p; G) = Z=p\Omega G which is the connecting homomorphism of the six term exact sequence obtained by tensoring G with the short exact sequence 0 ! Z=p ! Z=p 2 ! Z=p ! 0: The purpose of this paper to expand on this observation; indeed, our principal result will be that this is an algebraic ...