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 2-cubes 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 2-cubes 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)

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:

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 n-cubes operad. Our results are also valid for cosimplicial spectra. 1 Introduction. The little n-cubes 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 n-fold loop spaces. They showed that for any topological space Y the n-fold 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

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 2-cubes 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 2-cubes. 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

It is well known since Stasheff’s work that 1-fold 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 n-operads.

The inspiration for this thesis comes from mathematical physics, especially path integrals and the Chern-Simons 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

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