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15
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
Moduli space actions on the Hochschild CoChains of a Frobenius algebra I: Cell Operads
"... Abstract. This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co–chains of a Frobenius algebra. We also prove that a there is dg–PROP action of a version of Sullivan Chor ..."
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Abstract. This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co–chains of a Frobenius algebra. We also prove that a there is dg–PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild cochains of a Frobenius algebra. These actions lift to operadic correlation functions on the co–cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply–connected manifold. In this second part, we discretize the operadic and PROPic structures of the first part. We also introduce the notion of operadic correlation functions and use them in conjunction with operadic maps from the cell
Algebras of higher operads as enriched categories II
 In preparation
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the ..."
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of nglobular sets from any normalised (n + 1)operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)category is something like a category enriched in weak ncategories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.
The topological cyclic Deligne conjecture
 Algebr. Geom. Topol
"... Abstract. Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2discs operad. 1. ..."
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Abstract. Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2discs operad. 1.
The lattice path operad and Hochschild cochains
 CONTEMPORARY MATHEMATICS
"... We introduce two coloured operads in sets – the lattice path operad and a cyclic extension of it – closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an E2action (resp. framed E2action) on the Hochschild cochai ..."
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We introduce two coloured operads in sets – the lattice path operad and a cyclic extension of it – closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an E2action (resp. framed E2action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.
Multitensor lifting and strictly unital higher category theory
"... Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories ..."
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Abstract. In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result – the lifting theorem for multitensors – enables us to see the Gray tensor product of 2categories and the Crans tensor product of Gray categories as part of this framework. We define weak ncategories with strict units by means of a notion of reduced higher operad, using the theory of algebraic weak factorisation systems. Our second principal result is to establish a lax tensor product on the category of weak ncategories with strict units, so that enriched categories with respect to this tensor product are exactly weak (n+1)categories with strict units. 1.
Operads and Sheaf Cohomology
, 2004
"... I explain how to construct E∞ cochain algebras for computing classical sheaf cohomology and, in principle, hypercohomology, and I explain how not to construct E∞ cochain algebras for computing motivic cohomology. ..."
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I explain how to construct E∞ cochain algebras for computing classical sheaf cohomology and, in principle, hypercohomology, and I explain how not to construct E∞ cochain algebras for computing motivic cohomology.
MODULI SPACE ACTIONS ON THE HOCHSCHILD
, 2006
"... Abstract. This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co–chains of a Frobenius algebra. We also prove that a there is dg–PROP action of a version of Sullivan Chord ..."
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Abstract. This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co–chains of a Frobenius algebra. We also prove that a there is dg–PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild cochains of a Frobenius algebra. These actions lift to operadic correlation functions on the co–cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply–connected manifold. In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are that there is a cell model operad for the moduli space of genus g curves with n punctures and a tangent vector at each of these punctures and that there exists a CW complex whose chains are isomorphic to a certain type of Sullivan Chord diagrams and that they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.
arXiv version: fonts
"... pagination and layout may vary from GTM published version Poisson structures on the homology of the space of knots KEIICHI SAKAI In this article we study the Poisson algebra structure on the homology of the totalization of a fibrant cosimplicial space associated with an operad with multiplication. T ..."
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pagination and layout may vary from GTM published version Poisson structures on the homology of the space of knots KEIICHI SAKAI In this article we study the Poisson algebra structure on the homology of the totalization of a fibrant cosimplicial space associated with an operad with multiplication. This structure is given as the Browder operation induced by the action of little disks operad, which was found by McClure and Smith. We show that the Browder operation coincides with the Gerstenhaber bracket on the Hochschild homology, which appears as the E 2term of the homology spectral sequence constructed by Bousfield. In particular we consider a variant of the space of long knots in higher dimensional Euclidean space, and show that Sinha’s homology spectral sequence computes the Poisson algebra structure of the homology of the space. The Browder operation produces a homology class which does not directly correspond to chord diagrams. 55P48; 55P35 1