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11
A solution of Deligne’s Hochschild cohomology conjecture, Recent progress in homotopy theory
, 2000
"... ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an ..."
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Cited by 108 (3 self)
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ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action by an operad that is equivalent to the little 2cubes operad. 1 Introduction. Let us first recall some facts about the Hochschild cochain complex C ∗ (R) of an associative ring R. An element of C p (R) is a map of abelian groups x: R ⊗p → R. Hochschild [14] observed that there is a cup product in C ∗ (R); if x ∈ C p (R) and y ∈ C q (R)
Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 36 (7 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
Cosimplicial objects and little ncubes
 I. Amer. J. Math
"... In this paper we show that if a cosimplicial space has a certain kind of combinatorial structure then its total space has an action of an operad weakly equivalent to the little ncubes operad. Our results are also valid for cosimplicial spectra. 1 Introduction. The little ncubes operad Cn was intro ..."
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Cited by 25 (0 self)
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In this paper we show that if a cosimplicial space has a certain kind of combinatorial structure then its total space has an action of an operad weakly equivalent to the little ncubes operad. Our results are also valid for cosimplicial spectra. 1 Introduction. The little ncubes operad Cn was introduced by Boardman and Vogt in [6] (except that they used the terminology of theories rather than that of operads) as a tool for understanding nfold loop spaces. They showed that for any topological space Y the nfold loop space Ω n Y has an action of Cn. In the other direction, May showed in [19] that if Z is a space with
A solution of Deligne’s conjecture
"... ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an ..."
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Cited by 8 (0 self)
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ABSTRACT: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action of an operad equivalent to the little 2cubes. 1 Introduction. In order to explain Deligne’s question, let us first recall some facts about the Hochschild cochain complex C ∗ (R), where R is an associative ring. An element of C p (R) is a map of abelian groups
The combinatorics of iterated loop spaces
"... It is well known since Stasheff’s work that 1fold loop spaces can be described in terms of the existence of higher homotopies for associativity (coherence conditions) or equivalently as algebras of contractible nonsymmetric operads. The combinatorics of these higher homotopies is well understood an ..."
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Cited by 4 (0 self)
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It is well known since Stasheff’s work that 1fold loop spaces can be described in terms of the existence of higher homotopies for associativity (coherence conditions) or equivalently as algebras of contractible nonsymmetric operads. The combinatorics of these higher homotopies is well understood and is extremely useful. For n ≥ 2 the theory of symmetric operads encapsulated the corresponding higher homotopies, yet hid the combinatorics and it has remain a mystery for almost 40 years. However, the recent developments in many fields ranging from algebraic topology and algebraic geometry to mathematical physics and category theory show that this combinatorics in higher dimensions will be even more important than the one dimensional case. In this paper we are going to show that there exists a conceptual way to make these combinatorics explicit using the so called higher nonsymmetric noperads.
Ordinal subdivision and special pasting in quasicategories
"... Quasicategories are simplicial sets with properties generalising those of the nerve of a category. They model weak∞categories. Using a combinatorially defined ordinal subdivision, we examine composition rules for certain special pasting diagrams in quasicategories. The subdivision is of combinatori ..."
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Cited by 3 (0 self)
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Quasicategories are simplicial sets with properties generalising those of the nerve of a category. They model weak∞categories. Using a combinatorially defined ordinal subdivision, we examine composition rules for certain special pasting diagrams in quasicategories. The subdivision is of combinatorial interest in its own right and is linked with various combinatorial constructions. 1 Introduction. The most usual method of subdivision for a simplicial complex used in elementary algebraic and geometric topology is the barycentric subdivision. There is however another very well structured subdivision construction encountered from time to time. The basic geometric construction involves chopping up a
LATTICE GAUGE FIELD THEORY
, 2001
"... The inspiration for this thesis comes from mathematical physics, especially path integrals and the ChernSimons action. Path integrals were introduced by Feynman in late 1940’s and they have recently been applied to purely geometric problems. The work [33] of Edward Witten on the topological quantum ..."
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Cited by 1 (1 self)
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The inspiration for this thesis comes from mathematical physics, especially path integrals and the ChernSimons action. Path integrals were introduced by Feynman in late 1940’s and they have recently been applied to purely geometric problems. The work [33] of Edward Witten on the topological quantum field theory has been found very attractive by many enthusiastic
in monoidal categories
, 1995
"... We consider the theory of operads and their algebras in enriched category theory. We introduce the notion of simplicial A~cgraph and show that some important constructions of homotopy coherent category theory lead by a natural way to the use of such objects as the appropriate homotopy coherent coun ..."
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We consider the theory of operads and their algebras in enriched category theory. We introduce the notion of simplicial A~cgraph and show that some important constructions of homotopy coherent category theory lead by a natural way to the use of such objects as the appropriate homotopy coherent counterparts of the categories. @ 1998 Elsevier Science B.V. 1991 Math. Subj. Class.. " 18D20, 18D35, 18{330 Let A be a small simplicial category, and let F, G: A ~ K be two simplicial functors to a simplicial category K. Then we can consider, as the simplicial set of coherent natural transformations from F to G, the coherent end [8, 10, 12] (see Definition 6.2): Coh(F, G) =.fA K(F():) , G(2)).