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20
Spectral enrichments of model categories
"... We prove that every stable, presentable model category can be enriched in a natural way over symmetric spectra. As a consequence of the general theory, every object in such a model category has an associated homotopy endomorphism ring spectrum. Basic properties of these invariants are established. ..."
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Cited by 24 (5 self)
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We prove that every stable, presentable model category can be enriched in a natural way over symmetric spectra. As a consequence of the general theory, every object in such a model category has an associated homotopy endomorphism ring spectrum. Basic properties of these invariants are established.
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 14 (2 self)
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Abstract. We define the notion of an additive model category and prove that
Postnikov extensions for ring spectra
, 2006
"... We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum. ..."
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Cited by 13 (3 self)
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We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum.
Morita theory for derived categories: a bicategorical perspective
, 2008
"... Abstract. We present a bicategorical perspective on derived Morita theory for rings, DG algebras, and spectra. This perspective draws a connection between Morita theory and the bicategorical Yoneda Lemma, yielding a conceptual unification of Morita theory in derived and bicategorical contexts. This ..."
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Cited by 5 (1 self)
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Abstract. We present a bicategorical perspective on derived Morita theory for rings, DG algebras, and spectra. This perspective draws a connection between Morita theory and the bicategorical Yoneda Lemma, yielding a conceptual unification of Morita theory in derived and bicategorical contexts. This is motivated by study of Rickard’s theorem for derived equivalences of rings and of Morita theory for ring spectra, which we present in Sections 2 and 4. Along the way, we gain an understanding of the barriers to Morita theory for DG algebras and give a conceptual explanation for the counterexample of Dugger and Shipley. 1.
Local framings
"... Abstract. Framings provide a way to construct Quillen functors from simplicial sets to any given model category. A more structured setup studies stable frames giving Quillen functors from spectra to stable model categories. We will investigate how this is compatible with Bousfield localisation to ga ..."
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Cited by 4 (4 self)
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Abstract. Framings provide a way to construct Quillen functors from simplicial sets to any given model category. A more structured setup studies stable frames giving Quillen functors from spectra to stable model categories. We will investigate how this is compatible with Bousfield localisation to gain insight into the deeper structure of the stable homotopy category. We further show how these techniques relate to rigidity questions and how they can be used to study algebraic model
A curious example of triangulatedequivalent model categories which are not Quillen equivalent
 GEOM. TOPOL
, 2009
"... The paper gives a new proof that the model categories of stable modules for the rings Z=p 2 and Z=pŒ =. 2 / are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose ..."
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Cited by 4 (0 self)
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The paper gives a new proof that the model categories of stable modules for the rings Z=p 2 and Z=pŒ =. 2 / are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with nonisomorphic K–theories.
TOPOLOGICAL HOCHSCHILD AND CYCLIC HOMOLOGY FOR DIFFERENTIAL GRADED CATEGORIES
, 2008
"... We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several nontrivial natural transformations from algebraic Ktheory to THH(−). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equiva ..."
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Cited by 3 (2 self)
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We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several nontrivial natural transformations from algebraic Ktheory to THH(−). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zigzag of Quillen equivalences, to the homotopy theory of EilenbergMac Lane spectral categories. Finally, we show that over the rationals Q two dg categories are topological equivalent if and only if they are quasiequivalent.
Generalized spectral categories, topological Hochschild homology and trace maps
"... per we describe how to lift a model structure on the category of C–enriched categories to the category of Sp†.C;K/–enriched categories. This allow us to construct a (four step) zigzag of Quillen equivalences comparing dg categories to HZ–categories. As an application we obtain: (1) the invariance u ..."
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Cited by 3 (2 self)
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per we describe how to lift a model structure on the category of C–enriched categories to the category of Sp†.C;K/–enriched categories. This allow us to construct a (four step) zigzag of Quillen equivalences comparing dg categories to HZ–categories. As an application we obtain: (1) the invariance under weak equivalences of the topological Hochschild homology (THH) and topological cyclic homology (TC) of dg categories; (2) nontrivial natural transformations from algebraic K–theory to THH.
A curious example of two model categories and some associated differential graded algebras
, 2008
"... The paper gives a new proof that the model categories of stable modules for the rings Z/p² and Z/p[ɛ]/(ɛ²) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose ..."
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Cited by 2 (1 self)
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The paper gives a new proof that the model categories of stable modules for the rings Z/p² and Z/p[ɛ]/(ɛ²) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with nonisomorphic Ktheories.