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 Sym1 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 Sym1 is the first term of the infinite series of left adjoint functors Symn, 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 Symn(A).In this paper we consider some geometrical and homotopical aspects of the symmetrisation of n-operads. We follow Getzler and Jones and consider their decomposition of the Fulton-Macpherson operad of compactified real configuration spaces. We construct an n-operadic counterpart of this compactification which we call the Getzler-Jones operad. We study the properties of Getzler-Jones operad and find that it is contractible and cofibrant in an appropriate model category. The symmetrisation of the Getzler-Jones operad turns out to be exactly the operad of Fulton and Macpherson. These results should be considered as an extension of Stasheff's theory of 1-fold loop spaces to n-fold loop spaces n ≥ 2. We also show that a space X with an action of a contractible 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 prove analogous results in this case.

Locally constant $n$-operads as higher braided operads

“…We now recall the definition of pruned .n 1/-terminal n-operad [1]. Since we do not need other types of n-operads in this paper we will call them simply n-operads.…”

“…Let J n .k/ be the Milgram poset of all possible n-ordinal structures on the set f0; : : : ; k 1g [1]. The group S k acts on J n .k/ and the quotient J n .k/=S k is isomorphic to Q n .k/.…”

“…OEk be a morphism in s and let 0 Ä i Ä k. Then the preimage 1 .i/ has a linear order induced from OEn. Hence, there exists a unique object OEn i 2 s and a unique order-preserving bijection OEn i !…”

“…An example of an interesting locally constant n-operad in the model category of topological spaces which is not a quasisymmetric n-operad is the Getzler-Jones n-operad GJ n constructed in [1] for all n < 1. One can also construct an 1-version…”

“…We give a sketch of the proof. A detailed discussion can be found in [1], [3]. Consider the configuration space of ordered k-points in R n :…”