Results 1  10
of
27
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. ..."
Abstract

Cited by 63 (3 self)
 Add to MetaCart
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.
Equivalences of monoidal model categories
 Algebr. Geom. Topol
, 2002
"... Abstract: We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [ ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
Abstract: We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [SS00]. As an application we extend the DoldKan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [SS] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra. 1.
Integral transforms and Drinfeld centers in derived algebraic geometry
"... Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derive ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derived stacks which we call stacks with air. The class of stacks with air includes in particular all quasicompact, separated derived schemes and (in characteristic zero) all quotients of quasiprojective or smooth derived schemes by affine algebraic groups, and is closed under derived fiber products. We show that the (enriched) derived categories of quasicoherent sheaves on stacks with air behave well under algebraic and geometric operations. Namely, we identify the derived category of a fiber product with the tensor product of the derived categories of the factors. We also identify functors between derived categories of sheaves with integral transforms (providing a generalization of a theorem of Toën [To1] for ordinary schemes over a ring). As a first application, for a stack Y with air, we calculate the Drinfeld center (or synonymously,
Localization theorems in topological Hochschild homology and topological cyclic homology
, 2008
"... We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of ThomasonTrobaugh in Ktheory. We also deduce versions of Thomason’s blowup formula and the projective bundle formula for THH and TC.
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 ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
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
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. ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
We give a functorial construction of kinvariants for ring spectra, and use these to classify extensions in the Postnikov tower of a ring spectrum.
ENRICHED MODEL CATEGORIES AND AN APPLICATION TO ADDITIVE ENDOMORPHISM SPECTRA
"... Abstract. We define the notion of an additive model category and prove that ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
Abstract. We define the notion of an additive model category and prove that
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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.