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Notes on A∞algebras, A∞categories and noncommutative geometry, Homological mirror symmetry
 Lecture Notes in Phys
, 2009
"... 1.1 A∞algebras as spaces........................ 2 ..."
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1.1 A∞algebras as spaces........................ 2
Tamarkin’s proof of Kontsevich formality theorem
 Forum Math
"... 1.1. This is an extended version of lectures given at Luminy colloquium “Quantification par déformation ” held at December, 1999. In this note we explain Tamarkin’s proof [T] of the following affine algebraic version of Kontsevich’s formality theorem. ..."
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1.1. This is an extended version of lectures given at Luminy colloquium “Quantification par déformation ” held at December, 1999. In this note we explain Tamarkin’s proof [T] of the following affine algebraic version of Kontsevich’s formality theorem.
The EckmannHilton argument, higher operads and Enspaces, available at http://www.ics.mq.edu.au
 mbatanin/papers.html of Homotopy and Related Structures
"... The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of ..."
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The classical EckmannHilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2category, then its Homset is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we extend this argument to arbitrary dimension. We demonstrate that for an noperad A in the author’s sense there exists a symmetric operad S n (A) called the nfold suspension of A such that the
The BoardmanVogt resolution of operads in monoidal model categories, in preparation
"... Abstract. We extend the Wconstruction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for wellpointed Σcofibrant operads. The standard simplicial resolution of Godement as well as the cobarbar chain ..."
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Abstract. We extend the Wconstruction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for wellpointed Σcofibrant operads. The standard simplicial resolution of Godement as well as the cobarbar chain resolution are shown to be particular instances of this generalised Wconstruction.
Deformation theory of representations of prop(erad)s I
"... 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 element ..."
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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.
A proof of a cyclic version of Deligne’s conjecture via cacti
, 2004
"... Abstract. We generalize our results on Deligne’s conjecture to prove the statement that the normalized Hochschild co–chains of a finite–dimensional associative algebra with a non–degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which ..."
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Abstract. We generalize our results on Deligne’s conjecture to prove the statement that the normalized Hochschild co–chains of a finite–dimensional associative algebra with a non–degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in characteristic zero they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber’s bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the little discs operad in terms of spineless cacti with natural chain models for (S 1) ×n adapted to cacti. 1.
The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal
 Jour. Noncommutative Geom
, 2007
"... The solution of Deligne’s conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasiisomorphisms of homo ..."
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The solution of Deligne’s conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasiisomorphisms of homotopy Gerstenhaber algebras between the Hochschild cochain complex C • (A) of a regular commutative algebra A over a field K of characteristic zero and the Gerstenhaber algebra of multiderivations of A. Unlike the original approach of the second author based on the computation of obstructions our method allows us to avoid the bulky GelfandFuchs trick and prove the formality of the homotopy Gerstenhaber algebra structure on the sheaf of polydifferential operators on a smooth algebraic variety, a complex manifold, and a smooth real manifold.
On the cyclic Deligne conjecture
 J. Pure Appl. Algebra
"... Abstract. Let A be a finite dimensional, unital, associative algebra with nondegenerate invariant inner product. In this note, we give an explicit description of an action of chains of cyclic Sullivan chord diagrams on the reduced Hochschild cochain complex of A. This paper is prepared as notes for ..."
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Abstract. Let A be a finite dimensional, unital, associative algebra with nondegenerate invariant inner product. In this note, we give an explicit description of an action of chains of cyclic Sullivan chord diagrams on the reduced Hochschild cochain complex of A. This paper is prepared as notes for a talk at the AMS meeting in Lawrenceville in April 2004. It concerns structures on the reduced Hochschild cochain complex, mirroring those of String Topology. String Topology may be regarded as the study of the algebraic topology of the free loop space of a manifold. Chas and Sullivan, [1], [2], showed that the cohomology of the free loop space of a manifold has the structure of a BV algebra. Building on Sullivan’s later work [13], Cohen and Godin [3] showed that string topology operations give rise to a two dimensional positive boundary TQFT. This was achieved by looking at certain operations coming from so called Sullivan chord diagrams. The Hochschild cochain complex of the cochains of a (simply connected) manifold gives a model for the chains on the free loop space of that manifold. One expects