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183
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 190 (4 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.
DG quotients of DG categories
 J. ALGEBRA
, 2008
"... Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. ..."
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Cited by 152 (0 self)
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Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.
Introduction to Ainfinity algebras and modules
, 1999
"... These are slightly expanded notes of four introductory talks on ..."
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Cited by 117 (4 self)
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These are slightly expanded notes of four introductory talks on
Homotopical algebraic geometry. II. Geometric stacks and applications
, 2006
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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 ..."
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Cited by 88 (17 self)
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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,
Topological Open pBranes
, 2000
"... By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled ..."
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Cited by 45 (1 self)
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By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled theories. We show that, in general, topological open pbrane has a structure of (p + 1)algebra in the bulk, while a structure of palgebra in the boundary. The bulk/boundary correspondences are exactly as of the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of palgebras. It also imply that the algebras of quantum observables of (p − 1)brane are “close to ” the algebras of its classical observables as palgebras. We interpret above as deformation quantization of (p − 1)brane, generalizing the p = 1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to
Multivariable cochain operations and little ncubes
 J. Amer. Math. Soc
"... Abstract. In this paper we construct a small E ∞ chain operad S which acts naturally on the normalized cochains S ∗ X of a topological space. We also construct, for each n, a suboperad Sn which is quasiisomorphic to the normalized singular chains of the little ncubes operad. The case n = 2 leads t ..."
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Cited by 41 (2 self)
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Abstract. In this paper we construct a small E ∞ chain operad S which acts naturally on the normalized cochains S ∗ X of a topological space. We also construct, for each n, a suboperad Sn which is quasiisomorphic to the normalized singular chains of the little ncubes operad. The case n = 2 leads to a substantial simplification of our earlier proof of Deligne’s Hochschild cohomology conjecture. 1. Introduction. This paper has two goals. The first (see Theorem 2.15 and Remark 2.16(a)) is to construct a small E ∞ chain operad S which acts naturally on the normalized cochains S∗X of a topological space X. This is of interest in view of a theorem of Mandell [15, page 44] which states that if O is any E ∞ chain operad over Fp (the algebraic closure of the field with