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25
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.
The homotopy theory of dgcategories and derived Morita Theory
, 2006
"... The main purpose of this work is to study the homotopy theory of dgcategories up to quasiequivalences. Our main result is a description of the mapping spaces between two dgcategories C and D in terms of the nerve of a certain category of (C, D)bimodules. We also prove that the homotopy category ..."
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Cited by 61 (8 self)
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The main purpose of this work is to study the homotopy theory of dgcategories up to quasiequivalences. Our main result is a description of the mapping spaces between two dgcategories C and D in terms of the nerve of a certain category of (C, D)bimodules. We also prove that the homotopy category Ho(dg −Cat) possesses internal Hom’s relative to the (derived) tensor product of dgcategories. We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dgcategories of modules over two dgcategories C and D as the dgcategory of (C, D)bimodules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the classifying space of dgcategories (i.e. the nerve of the category of dgcategories and quasiequivalences between them). The second application is the existence of a good theory of localization for dgcategories, defined in terms of a natural universal property. Our last application states that the dgcategory of (continuous) morphisms between the dgcategories of quasicoherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasiequivalent
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
Topological equivalences for differential graded algebras
 Adv. Math
, 2006
"... Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are ..."
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Cited by 14 (6 self)
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Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents
On derived equivalences of categories of sheaves over finite posets
 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
"... Abstract. A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D b (X) the bounded derived category of sheaves of finite dimensional kvector spaces over X. Two posets X and Y are said to be derived equivalent if D b (X) and D b (Y) are equivalent as tria ..."
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Cited by 11 (7 self)
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Abstract. A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D b (X) the bounded derived category of sheaves of finite dimensional kvector spaces over X. Two posets X and Y are said to be derived equivalent if D b (X) and D b (Y) are equivalent as triangulated categories. We give explicit combinatorial properties of X which are invariant under derived equivalence, among them are the number of points, the Zcongruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence. For any closed subset Y ⊆ X, we construct a strongly exceptional collection in D b (X) and use it to show an equivalence D b (X) ≃ D b (A) for a finite dimensional algebra A (depending on Y). We give conditions on X and Y under which A becomes an incidence algebra of a poset. We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite S op. This construction produces many new derived equivalences of posets and generalizes other well known ones. As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands. 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
Homotopy theory of spectral categories
, 2008
"... Abstract. We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories. Following Toën’s work on DG categories, we describe the mapping space between two spectral categories A ..."
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Cited by 5 (1 self)
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Abstract. We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories. Following Toën’s work on DG categories, we describe the mapping space between two spectral categories A and B in terms of the nerve of a certain category of ABbimodules, and show that the homotopy category of spectral categories obtained admits internal Hom’s relative to the derived smash product. As an application, we obtain a simple conceptual characterization of topological
Almost Dsplit sequences and derived equivalences. preprint
"... In this paper, we introduce almost Dsplit sequences and establish an elementary but somewhat surprising connection between derived equivalences and AuslanderReiten sequences via BBtilting modules. In particular, we obtain derived equivalences from AuslanderReiten sequences (or nalmost split seq ..."
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Cited by 4 (2 self)
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In this paper, we introduce almost Dsplit sequences and establish an elementary but somewhat surprising connection between derived equivalences and AuslanderReiten sequences via BBtilting modules. In particular, we obtain derived equivalences from AuslanderReiten sequences (or nalmost split sequences), and AuslanderReiten triangles. 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.
A curious example of two model categories and some associated differential graded algebras, in preparation
"... Abstract. The paper gives a new proof that the model categories of stable modules for the rings Z/p 2 and Z/p[ɛ]/(ɛ 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 ..."
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Cited by 2 (1 self)
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Abstract. The paper gives a new proof that the model categories of stable modules for the rings Z/p 2 and Z/p[ɛ]/(ɛ 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 Ktheories. Contents