Abstract:
It is classical observation that the functor from symmetric operads to nonsymmetric operads which forgets the symmetric groups action has a left adjoint. This left adjoint, called symmetrisation, is simply a multiplication on the so called permutation operad. In my lecture I will show how to generalise this adjunction for an adjunction between symmetric and n-operads for any n (the classical case corresponds to n=1). I will relate this adjunction to the combinatorics of compactified configuration spaces and I will generalise Stasheff's combinatorial discription of one fold loop spaces to higher dimensions. If time permits I will give a conceptual proof of Deligne's hypothesis on Hochschild cochains (due to Tamarkin) as another application of the categorical formalism obtained.