Symmetrisation of $n$-operads and compactification of real configuration spaces

Abstract

It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint $Sym_1$ given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author's theory of higher operads, the nonsymmmetric operads are 1-operads and $Sym_1$ is the first term of the infinite series of left adjoint functors $Sym_n,$ called symmetrisation functors, from $n$-operads to symmetric operads with the property that the category of one object, one arrow, . . ., one $(n-1)$-arrow algebras of an $n$-operad $A$ is isomorphic to the category of algebras of $Sym_n(A)$. In this paper we consider some geometrical and homotopical aspects of the symmetrisation of $n$-operads. We construct an $n$-operadic analogue of Fulton-Macpherson operad and show that its symmetrisation is exactly the operad of Fulton and Macpherson. This implies that a space $X$ with an action of a ontractible $n$-operad has a natural structure of an algebra over an operad weakly equivalent to the little $n$-disks operad. A similar result holds for chain operads. These results generalise the classical Eckman-Hilton argument to arbitrary dimension. Finally, we apply the techniques to the Swiss Cheese type operads introduced by Voronov and get analogous results in this case.