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Symmetrisation of \(n\)-operads and compactification of real configuration spaces. (English) Zbl 1146.18003

This is the second paper in a series on higher operads. The first [M. A. Batanin, “The Eckmann-Hilton argument and higher operads”, Adv. Math. 217, No. 1, 334–385 (2008; Zbl 1138.18003)] introduced tools for going back and forth between \(n\)-operads in the sense of “Monoidal globular categories as a natural environment for the theory of weak \(n\)-categories” [M. A. Batanin, Adv. Math. 136, No. 1, 39–103 (1998; Zbl 0912.18006)] and classical symmetric operads. A symmetrisation functor \(\text{Sym}_n\) from \(n\)-operads to symmetric operads was defined, left adjoint to a desymmetrisation functor in the opposite direction. The functor \(\text{Sym}_n\) has the property that, for an \(n\)-operad \(A\), the category of algebras of \(\text{Sym}_n(A)\) is isomorphic to the category of one object, one arrow,…, one \((n-1)\)-arrow algebras of \(A\).
A technical point in this paper is that the author restricts to so-called pruned \((n-1)\)-terminal \(n\)-operads, making the combinatorics easier and still allowing the main applications. It is shown that a space with an action of a contractible pruned \((n-1)\)-terminal \(n\)-operad has an action of an \(E_n\)-operad.
An \(n\)-operadic version of the Getzler-Jones decomposition of the Fulton-Macpherson operad of compactified real configuration spaces is described. This was first discussed in a 1994 preprint of Getzler and Jones, before the machinery of \(n\)-operads was available. The paper uses an explicit description of the compactification of the configuration space of \(k\) points in \(\mathbb{R}^n\) from M. Kontsevich and Y. Soibelman [“Affine structures and non-Archimedean analytic spaces”, Prog. Math. 244, 321–385 (2006; Zbl 1114.14027)] and D. P. Sinha [“Manifold-theoretic compactifications of configuration spaces”, Sel. Math., New Ser. 10, No. 3, 391–428 (2004; Zbl 1061.55013)]. There are various closely related operads built out of the collection of these configuration spaces (as \(k\) varies), including the Fulton-Macpherson operad \(\mathbf{fm}^n\), a reduced symmetric operad weakly equivalent to the little \(n\)-cubes operad. The configuration spaces have decompositions into Fox-Neuwirth cells and there are corresponding Getzler-Jones cells \(\text{Mod}^n_T\), indexed by pruned \(n\)-trees, giving a decomposition of \(\mathbf{fm}^n\). However, as observed by Tamarkin, this decomposition does not give a cellular structure compatible with the operad structure.
The Getzler-Jones \(n\)-operad is defined as follows. First take the free reduced \(n\)-operad generated by the \(n\)-collection \(\text{Mod}^n_{\bullet}\). It turns out that this maps via a continuous injection of operads to the desymmetrisation of \(\mathbf{fm}^n\) and its image under this map is the Getzler-Jones \(n\)-operad, \(\mathbf{GJ}^n\). It is proved that its symmetrisation is the Fulton-Macpherson operad: \(\text{Sym}_n(GJ^n)\cong \mathbf{fm}^n\).
Properties of the Getzler-Jones operad are studied and it is shown to be contractible. Model structures for \(n\)-operads are introduced, generalizing the cofibrantly generated model structure of Berger and Moerdijk for reduced symmetric operads [C. Berger and I. Moerdijk, “Axiomatic homotopy theory for operads”, Comment. Math. Helv. 78, No. 4, 805–831 (2003; Zbl 1041.18011)]. Weak equivalences (fibrations, respectively) between topological or simplicial \(n\)-operads are operadic maps which are termwise weak equivalences (fibrations, respectively) in topological spaces or simplicial sets. The Getzler-Jones \(n\)-operad is shown to be cofibrant in this model structure.
Study of the combinatorics of the Getzler-Jones \(n\)-operad gives rise to a generalisation of Stasheff’s theory of \(A_{\infty}\)-spaces. The cellular structure of the operad means that its action can be described via an inductive process of extension of higher homotopies from the boundary of the closure \(K_T\) of a Getzler-Jones cell to its interior. The \(K_T\) are indexed by pruned \(n\)-trees and each is a manifold with corners homeomorphic to a ball of dimension \(E(T)-n-1\) where \(E(T)\) is the number of edges of the \(n\)-tree. These encode all the coherence conditions for \(E_n\)-spaces.
The final section covers higher analogues of Voronov’s Swiss-Cheese operads [A. A. Voronov, “The Swiss-Cheese operad”, Contemp. Math. 239, 365–373 (1999; Zbl 0946.55005)]. These are special coloured operads with two colours and they were introduced to provide finite-dimensional models of moduli spaces appearing in open-closed string theory.

MSC:

18D50 Operads (MSC2010)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
55P48 Loop space machines and operads in algebraic topology
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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References:

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