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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.
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.
A Cellular Nerve for Higher Categories
, 2002
"... ... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there ..."
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Cited by 21 (2 self)
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... categories. The associated cellular nerve of an ocategory extends the wellknown simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of Aalgebras for each ooperad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of Aalgebras (i.e. weak ocategories) is
Morita theory in abelian, derived and stable model categories, Structured ring spectra
 London Math. Soc. Lecture Note Ser
, 2004
"... These notes are based on lectures given at the Workshop on Structured ring spectra and ..."
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Cited by 19 (0 self)
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These notes are based on lectures given at the Workshop on Structured ring spectra and
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.
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
On Voevodsky’s algebraic Ktheory spectrum BGL
, 2007
"... Under a certain normalization assumption we prove that the P 1spectrum BGL of Voevodsky which represents algebraic Ktheory is unique over Spec(Z). Following an idea of Voevodsky, we equip the P 1spectrum BGL with the structure of a commutative ring P 1spectrum in the motivic stable homotopy cate ..."
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Cited by 13 (6 self)
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Under a certain normalization assumption we prove that the P 1spectrum BGL of Voevodsky which represents algebraic Ktheory is unique over Spec(Z). Following an idea of Voevodsky, we equip the P 1spectrum BGL with the structure of a commutative ring P 1spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over Spec(Z). For an arbitrary Noetherian base scheme S we pull this structure back to get a distinguished monoidal structure on BGL. 1
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.
ENRICHED MODEL CATEGORIES AND AN APPLICATION TO ADDITIVE ENDOMORPHISM SPECTRA
"... Abstract. We define the notion of an additive model category and prove that ..."
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Cited by 9 (3 self)
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Abstract. We define the notion of an additive model category and prove that
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.