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19
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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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.
Representations up to homotopy of Lie algebroids
, 2008
"... Abstract We introduce, based on Quillen's superconnections, the notion of representation up tohomotopy in the general context of Lie algebroids. We use the resulting framework to make ..."
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Cited by 4 (2 self)
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Abstract We introduce, based on Quillen's superconnections, the notion of representation up tohomotopy in the general context of Lie algebroids. We use the resulting framework to make
Units of ring spectra and Thom spectra
"... Abstract. We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E ∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor To a map of spectra Σ ∞ + Ω ∞ : ho(c ..."
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Abstract. We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E ∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor To a map of spectra Σ ∞ + Ω ∞ : ho(connective spectra) → ho(E ∞ ring spectra). f: b → bgl1A, we associate an E ∞ Aalgebra Thom spectrum Mf, which admits an E ∞ Aalgebra map to R if and only if the composition b → bgl1A → bgl1R is null; the classical case developed by [MQRT77] arises when A is the sphere spectrum. We develop the analogous theory for A ∞ ring spectra. If A is an A ∞ ring spectrum, then to a map of spaces f: B → BGL1A we associate an Amodule Thom spectrum Mf, which admits an Rorientation if and only if
GALE DUALITY AND KOSZUL DUALITY
"... ABSTRACT. Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finitedimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles ..."
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Cited by 3 (2 self)
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ABSTRACT. Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finitedimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of
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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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.
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 2 (0 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.
THE ALGEBRA OF SECONDARY HOMOTOPY OPERATIONS IN RING SPECTRA
, 2007
"... Abstract. The primary algebraic model of a ring spectrum R is the ring π∗R of homotopy groups. We introduce the secondary model π∗,∗R which has the structure of a secondary analogue of a ring. The homology of π∗,∗R is π∗R and triple Massey products in π∗,∗R coincide with Toda brackets in π∗R. We als ..."
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Abstract. The primary algebraic model of a ring spectrum R is the ring π∗R of homotopy groups. We introduce the secondary model π∗,∗R which has the structure of a secondary analogue of a ring. The homology of π∗,∗R is π∗R and triple Massey products in π∗,∗R coincide with Toda brackets in π∗R. We also describe the secondary model of a commutative ring spectrum Q from which we derive the cupone square operation in π∗Q. As an application we obtain for each ring spectrum R new derivations of the ring π∗R. Contents
SECONDARY ALGEBRAS ASSOCIATED TO RING SPECTRA
, 2006
"... Abstract. Homotopy groups of a connective ring spectrum R form angraded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R ..."
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Abstract. Homotopy groups of a connective ring spectrum R form angraded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R, such as Toda brackets, and the first Postnikov invariant of R as a ring spectrum. Moreover, π∗,∗R represents a cohomology class in the third Mac Lane cohomology of the algebra π∗R. If R is commutative then π∗,∗R has an E∞structure and encodes the cupone squares in π∗R. Contents
On the derived category of an algebra over an operad, in preparation
"... Dedicated to Mamuka Jibladze on the occasion of his 50th birthday Abstract. We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad. ..."
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Dedicated to Mamuka Jibladze on the occasion of his 50th birthday Abstract. We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.
TODA BRACKETS AND CUPONE SQUARES FOR RING SPECTRA
, 2006
"... Abstract. In this paper we prove the laws of Toda brackets on the homotopy groups of a connective ring spectrum and the laws of the cupone square in the homotopy groups of a commutative connective ring spectrum. Contents ..."
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Abstract. In this paper we prove the laws of Toda brackets on the homotopy groups of a connective ring spectrum and the laws of the cupone square in the homotopy groups of a commutative connective ring spectrum. Contents