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57
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 66 (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.
Axiomatic Homotopy Theory for Operads
 Comment. Math. Helv
, 2002
"... We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced. ..."
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Cited by 54 (7 self)
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We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.
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 29 (1 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
A Koszul duality for props
 Trans. of Amer. Math. Soc
"... Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props. ..."
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Cited by 22 (4 self)
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Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.
A bordism approach to string topology
"... Abstract. Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology constructed by M. Chas and D. Sullivan, V. Godin and R. Cohen. We generalize some of these operations to spaces of maps from a sphere to a comp ..."
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Cited by 19 (1 self)
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Abstract. Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology constructed by M. Chas and D. Sullivan, V. Godin and R. Cohen. We generalize some of these operations to spaces of maps from a sphere to a compact manifold. 1.
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S I
"... Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCarta ..."
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Cited by 9 (4 self)
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Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.
Toën, Simplicial localization of monoidal structures and a nonlinear version of Deligne’s conjecture
 Compos. Math
"... Abstract. We show that if (M, ⊗, I) is a monoidal model category then REnd ..."
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Cited by 7 (1 self)
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Abstract. We show that if (M, ⊗, I) is a monoidal model category then REnd
What do dgcategories form
 Compositio Math
"... It is well known that categories form a 2category: 1arrows are functors and 2arrows are their natural transformations. In a similar way, dgcategories also form a 2category: 1arrows A → B are dgfunctors; given a pair of dgfunctors F,G: A → B one can define a complex of their natural transform ..."
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Cited by 6 (1 self)
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It is well known that categories form a 2category: 1arrows are functors and 2arrows are their natural transformations. In a similar way, dgcategories also form a 2category: 1arrows A → B are dgfunctors; given a pair of dgfunctors F,G: A → B one can define a complex of their natural transformations
Iterated bar complexes of Einfinity algebras and homology theories
, 2008
"... We proved in a previous article that the bar complex of an E ∞algebra inherits a natural E ∞algebra structure. As a consequence, a welldefined iterated bar construction B n (A) can be associated to any algebra over an E ∞operad. In the case of a commutative algebra A, our iterated bar constructi ..."
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We proved in a previous article that the bar complex of an E ∞algebra inherits a natural E ∞algebra structure. As a consequence, a welldefined iterated bar construction B n (A) can be associated to any algebra over an E ∞operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E ∞algebras. We use this effective definition to prove that the nfold bar construction admits an extension to categories of algebras over Enoperads. Then we prove that the nfold bar complex determines the homology theory associated to the category of algebras over an Enoperad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γhomology with trivial coefficients.
ON THE COBAR CONSTRUCTION OF A BIALGEBRA
, 2004
"... We show that the cobar construction of a DGbialgebra is a homotopy Galgebra. This implies that the bar construction of this cobar is a DGbialgebra as well. ..."
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Cited by 3 (3 self)
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We show that the cobar construction of a DGbialgebra is a homotopy Galgebra. This implies that the bar construction of this cobar is a DGbialgebra as well.