Symmetrisation of n-operads and compactification of real configuration spaces

By Michael Batanin. 2006

52 citations recorded by the source corpus.

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 nonsymmetric 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 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.

Subfields: Algebra and Number Theory, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/9aee547fb9f3bbb99557d82b2f69e6d6df470aea

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In Algebra and Number Theory

1. Martin Markl
2. Michael Batanin
3. C. Berger
4. Benoit Fresse
5. Ieke Moerdijk
6. L. Pereira
7. P. Bonventre
8. David White

In Mathematical Physics

1. Michael Batanin
2. Martin Markl
3. C. Berger
4. Benoit Fresse
5. Carlos Simpson
6. Ieke Moerdijk
7. D. Ara
8. L. Pereira

In Molecular Biology

1. C. Berger
2. Martin Markl

Citations in the corpus

• Homotopy theory for algebras over polynomial monads (2013, unknown): “We now recall the definition of a pruned (n − 1)-terminal n-operad [5].”
• Familial 2-functors and parametric right adjoints (2007, unknown): “The construction of operads which are universal among those which have a certain type of internal ! operad structure, has been shown in [Batanin, 2002] and [Batanin, 2006] to be fundamental to the theory of loop spaces.”
• Operadic categories and Duoidal Deligne's conjecture (2014, background): “They include categories like finite sets, finite ordinals, the categories of n-ordinals [4] and n-trees [3, 10], Barwick operator categories [2], ordered graphs and many other.”
• Iterated wreath product of the simplex category and iterated loop spaces (2005, result): “Batanin and Street [7] define a category Ωn which is closely related to Joyal’s category of finite combinatorial n-disks and open disk maps; however, Ωn contains in addition for each level-tree S of height < n degenerate level-trees ZkS for 0 < k ≤ n−ht(S), to the effect that those open disk maps wh”