## The symmetrisation of n-operads and compactification of real configuration spaces

Venue: | Adv. Math |

Citations: | 15 - 3 self |

### BibTeX

@ARTICLE{Batanin_thesymmetrisation,

author = {M. A. Batanin},

title = {The symmetrisation of n-operads and compactification of real configuration spaces},

journal = {Adv. Math},

year = {},

pages = {684--725}

}

### Years of Citing Articles

### OpenURL

### 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

### Citations

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Citation Context ... of R-chains of the little disc operad. Finally, Theorems 7.2 , 7.3 imply that we can obtain the full solution of the coherence problem of n-fold loop spaces in the spirit of Stasheff’s original work =-=[21, 22]-=- using cells KT for higher n-trees instead of associahedra. This was first claimed by Getzler and Jones but some doubts appeared since Tamarkin came up with his counterexample. Our Theorems 7.2 , 7.3 ... |

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Citation Context ...analogous theorem holds in the pruned case and in the reduced and unbased reduced case if V satisfies the assumptions of Theorem 5.3. Proof. It follows from the general properties of bar-construction =-=[17]-=- that the morphismn ρ is a deformation retraction in Coll (n−1) n . Hence, ρ is a trivial fibration of operads. It remains to prove that B(Fn, Fn, X) is a cofibrant n-operad. Let f : E → B be a trivia... |

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Citation Context ...t V be a model category which satisfies the conditions of Theorem 5.4, has a simplicial enrichment V S (−, −) satisfying 5.1, and which is a simplicial model category with respect to these structures =-=[11]-=-. Let X be an (n −1)-terminal n-operad in V with cofibrant underlying n-collection. Then the canonical operad morphism ρ : B(Fn, Fn, X) −→ X is a cofibrant replacement for X in the model category of (... |

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Citation Context ... space. An analogous result holds for the reduced operads in chain complexes. As we conjectured in [4] this should give a very natural proof of Deligne’s conjecture answering a question by Kontsevich =-=[13]-=- 1 . There are, however, other results in our paper which we believe are significant. First we observe that the desymetrisation of the operad of Fulton and Macpherson fm n of compactified real configu... |

112 | Deformations of algebras over operads and the Deligne conjecture. Conférence Moshé Flato
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Citation Context ..., however, other results in our paper which we believe are significant. First we observe that the desymetrisation of the operad of Fulton and Macpherson fm n of compactified real configuration spaces =-=[9, 14]-=- contains a contractible reduced (n − 1)-terminal n-operad which we call the Getzler-Jones operad GJ n . Actually, this operad was discovered by Getzler and Jones in their remarkable preprint [9]. The... |

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Citation Context ...truct the next manifold by gluing the outer hemisphere C(bl, rl, Rl) to the inner hemisphere of C(b0, r0, R0) in the place l. Example 7.2 The following example illustrates the proof. Here T = [5] ρ → =-=[3]-=- → [1] ρ(1) = ρ(2) = ρ(3) = 1 , ρ(4) = 2 , ρ(5) = 3 33and the Getzler-Jones cell corresponds to the map σ : T ′ → S of 2-trees S = [2] → [1] → [1] , σ(1) = σ(2) = 1 , σ(3) = σ(4) = 2. Obviously, this... |

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Citation Context ... S be an order preserving map of n-ordinals For an element i ∈ S the preimage f −1 (i) with its natural structure of an n-ordinal induced from T will be called a fiber of f over i. Following [10] and =-=[7]-=- we define a tree to be an isomorphism class of finite connected acyclic graphs with a marked vertex v0 called the root. A vertex which is not the root vertex and having valency more than 2 is called ... |

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Citation Context ..., however, other results in our paper which we believe are significant. First we observe that the desymetrisation of the operad of Fulton and Macpherson fm n of compactified real configuration spaces =-=[9, 14]-=- contains a contractible reduced (n − 1)-terminal n-operad which we call the Getzler-Jones operad GJ n . Actually, this operad was discovered by Getzler and Jones in their remarkable preprint [9]. The... |

23 | Manifold-theoretic compactifications of configuration spaces
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Citation Context ... Getzler-Jones decomposition The operadic structure on compactified moduli space of configurations of points in ℜ n was first observed by Getzler and Jones in [9]. Here we use an explicit approach of =-=[14, 20]-=- to describe this compactification. Let mod n [k] be the quotient of the configuration space Confk(ℜ n ) = {(x1, . . .,xk) ∈ (ℜ n ) k | xi ̸= xj if i ̸= j } with respect to the obvious action of the (... |

22 |
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Citation Context ...implicial simplicial set consisting of standard simplices. Since ∆ ⋆ is cofibrant in the Reedy model structure [11] it remains to show that f ⋆ is a trivial fibration. We follow a method developed in =-=[2]-=-. We have to prove that in the diagram 23Oper S n (Fi+1 n X, E) ❳ ❅ ❳❳❳ ωi ❳❳❳❳❳ ❅❘ ✲ Wi Mi(OperS n (F⋆n X, E)) ❆ ❆❆❆❆❆❆❯ OperS n(F i+1 ❄ n X, B) ✲ Mi(OperS n (F⋆ Mif ❄ nX, B)) ⋆ the canonical map ω... |

21 |
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Citation Context ... dijkdikj = dijkdikldilj = dijkdjkidkij = 1. Proof. The proof can be obtained using the explicit formulas for the operadic multiplication in fm n from [16] or techniques from [20] . ♣ Following Joyal =-=[12]-=- we give a definition of a generalised n-tree. Definition 6.1 A generalised n-tree X is a chain of partially ordered sets and functions n ρn−1 n−1 ρn−2 R −→ R −→ . . . ρ1 1 ρ0 −→ R −→ R 0 = 1 such tha... |

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Citation Context ... of the operad fm n . It turned out, however, that Getzler-Jones subdivision does not give a cellular structure compatible with the operadic structure of fm n as was first observed by D.Tamarkin (see =-=[25]-=- for an explanation of Tamarkin’s counterexample.) This counterexample implied considerable technical difficulties in the proof of Deligne’s conjecture. The second implicit appearance of GJ n was in t... |

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Citation Context ...le and the following relations between dijk [20]: dijkdikj = dijkdikldilj = dijkdjkidkij = 1. Proof. The proof can be obtained using the explicit formulas for the operadic multiplication in fm n from =-=[16]-=- or techniques from [20] . ♣ Following Joyal [12] we give a definition of a generalised n-tree. Definition 6.1 A generalised n-tree X is a chain of partially ordered sets and functions n ρn−1 n−1 ρn−2... |

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3 | The Eckman-Hilton argument and higher operads, preprint
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Citation Context ...operads and their symmetrisation 43 1 Introduction This is the second paper in a sequence of papers devoted to the relations between higher categories and n-fold loop space theory. In the first paper =-=[4]-=- we developed the necessary categorical techniques which allow to go back and forth between n-operads and classical symmetric operads. The main goal of this paper is to clarify the geometric and homot... |

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2 |
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Citation Context ... we construct the next manifold by gluing the outer hemisphere C(bl, rl, Rl) to the inner hemisphere of C(b0, r0, R0) in the place l. Example 7.2 The following example illustrates the proof. Here T = =-=[5]-=- ρ → [3] → [1] ρ(1) = ρ(2) = ρ(3) = 1 , ρ(4) = 2 , ρ(5) = 3 33and the Getzler-Jones cell corresponds to the map σ : T ′ → S of 2-trees S = [2] → [1] → [1] , σ(1) = σ(2) = 1 , σ(3) = σ(4) = 2. Obvious... |

2 |
What do DG-categories form? , preprint
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Citation Context ...ions. In the particular case of a DG-category with one object and transformations of the identity functor, we obtain the Hochschild complex of an associative algebra and then we apply our Theorem 8.7 =-=[23]-=-. 3operads in the model category theoretic sense, and in particular it is a cofibrant contractible operad. The term unbased here means that we forget about nullary operations of our operads. There is... |

1 |
Configuration spaces with summable labels,Progr
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Citation Context ...hism Symn(RH n ) ≃ rh n . This isomorphism induces an isomorphism of nerves N(rh n ) → Symn(N(RH n )). 17We also want to introduce an unbased version of reduced operads (we follow the terminology of =-=[19]-=-). These are reduced symmetric operads without nullary operations and with A1 = I. Notice, that we do not require A0 to be ∅ but simply forget about the 0-arity of our operads. This is, of course, a l... |

1 |
Swiss-Cheese operads, letter to M.Batanin, 2006. 47 Voronov A., Homotopy Gerstenhaber algebras, Confrence Mosh Flato
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Citation Context ...iss Cheese operad from n-operadic point of view and for sending me a preliminary version of his papers concerning action of Swiss-Cheese operads on associative algebras and their Hochschild complexes =-=[24]-=- 4objects for us here are (n − 1)-terminal n-algebras of our n-operads. Roughly speaking such an algebra is an object X of our basic symmetric monoidal category V together with an action AT ⊗X ⊗k → X... |

1 |
The Swiss-cheese operad, Contemp
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Citation Context ...ave full source and target operations. We will consider this subject in a forthcoming paper. Finally, in the last section of our paper we apply our techniques to the case of Swiss Cheese type operads =-=[26]-=-. The advantage of our categorical methods is that we have nothing to prove here once we put the right definitions of our main categories and functors in place. We deduce immediately a symmetrisation ... |

1 |
What do DG-categories form? , to appear in Compositio Math., available at math.CT/0606553 . 47 Tamarkin D.E., Swiss-Cheese operads, letter to M.Batanin
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Citation Context ...ions. In the particular case of a DG-category with one object and transformations of the identity functor, we obtain the Hochschild complex of an associative algebra and then we apply our Theorem 8.7 =-=[23]-=-. 3n-operads in the model category theoretic sense, and in particular it is a cofibrant contractible operad. The term unbased here means that we forget about nullary operations of our operads. There ... |