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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 78 (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.
On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 63 (3 self)
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
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.
Modern foundations of stable homotopy theory. Handbook of Algebraic Topology, edited by
, 1995
"... 2. Smash products and twisted halfsmash products 11 ..."
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Cited by 22 (7 self)
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2. Smash products and twisted halfsmash products 11
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
Commutative algebra in stable homotopy theory and a completion theorem
 Math. Res. Letters
, 1994
"... Abstract. We construct a category of spectra that has all limits and colimits and also has a strictly associative and commutative smash product. This provides the ground category for a new theory of structured ring and module spectra that allows the wholesale importation of techniques of commutative ..."
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Cited by 13 (9 self)
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Abstract. We construct a category of spectra that has all limits and colimits and also has a strictly associative and commutative smash product. This provides the ground category for a new theory of structured ring and module spectra that allows the wholesale importation of techniques of commutative algebra into stable homotopy theory. Applications include new constructions of basic spectra, new generalized universal coefficient and Künneth spectral sequences, and a new construction of topological Hochschild homology. The theory works equivariantly, where it allows the construction of equivariant versions of BrownPeterson, Morava Ktheory, and other module spectra over MU. Via a topological realization of “local homology and cohomology groups”, the general theory leads to a completion theorem for the computation of M∗(BG) and M ∗ (BG) in terms of equivariant cobordism groups, where M is MU, BP, k(n), K(n), or any other module spectrum over MU. (The reader most interested in the equivariant applications may wish to read the last section first.)
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 6 (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.
Morita theory in stable homotopy theory
, 2004
"... We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ fr ..."
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Cited by 3 (2 self)
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We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.
The norder of algebraic triangulated categories
"... Abstract. We quantify certain features of algebraic triangulated categories using the ‘norder’, an invariant that measures how strongly n annihilates objects of the form Y/n. We show that the norder of an algebraic triangulated category is infinite, and that the porder of the plocal stable homoto ..."
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Cited by 1 (1 self)
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Abstract. We quantify certain features of algebraic triangulated categories using the ‘norder’, an invariant that measures how strongly n annihilates objects of the form Y/n. We show that the norder of an algebraic triangulated category is infinite, and that the porder of the plocal stable homotopy category is exactly p − 1, for any prime p. In particular, the plocal stable homotopy category is not algebraic.
Why H Zalgebra Spectra are Differential Graded Algebras?
"... I would like to thank John Edward Harper for his assistance in this project. I am also grateful to Kathryn Hess Bellwald for her support and for suggesting the target article. My special thanks to the expert, Paul Turner, who will read this report. I appreciate the courage of Joseph Stupey, who has ..."
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I would like to thank John Edward Harper for his assistance in this project. I am also grateful to Kathryn Hess Bellwald for her support and for suggesting the target article. My special thanks to the expert, Paul Turner, who will read this report. I appreciate the courage of Joseph Stupey, who has watched all the Greek episodes. Finally, I would like to say thank you to Peter Jossen for his help and