The operad of finite labeled tournaments

We study the operad of finite labeled tournaments. We describe the structure of suboperads of this operad generated by simple tournaments. We prove that a suboperad generated by a tournament with two vertices (i.e., the operad of finite linearly ordered sets) is isomorphic to the operad of symmetric groups, and a suboperad generated by a simple tournament with more that two vertices is isomorphic to the quotient operad of the free operad with respect to a certain congruence. We obtain this congruence explicitly.     

A proof of the existence of Batanin's initial operad

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.     

On Hopf algebra structures over free operads

The operad Lie can be constructed as the operad of primitives PrimAs from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore. We replace the operad As by families of free operads P, which include the operad Mag freely generated by a non-commutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincaré-Birkhoff-Witt type theorems and collect information about the operads PrimP, e.g., in terms of characteristic functions.     

Higher genus surface operad detects infinite loop spaces

Abstract. The operad studied in conformal field theory and introduced ten years ago by G. Segal [S] is built out of moduli spaces of Riemann surfaces. We show here that this operad which at first sight is a double loop space operad is indeed an infinite loop space operad. This leads to a new proof of the fact that the classifying space of the stable mapping class group , is an infinite loop space after plus construction [T2]. This new approach has various advantages. In particular, the infinite loop space structure is more explicid. Received: 21 September 1998 / Revised: 30 August 1999 / Published online: 8 May 2000     

Une décomposition prismatique de l'opérade de Barratt–EcclesA prismatic decomposition of the Barratt–Eccles operad

The Barratt–Eccles operad is a simplicial operad formed by the classical homogeneous bar construction of symmetric groups. We prove that these simplicial sets decompose as unions of prisms indexed by surjections. We observe that the cellular complexes given by this prismatic structure are nothing but the components of the surjection operad (the operad introduced by J. McClure and J. Smith in their work on the Deligne conjecture). To cite this article: C. Berger, B. Fresse, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 365–370.     

Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen-Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.     

Cogroups in algebras over an operad are free algebras

Let be an operad defined over a field of characteristic zero. Let R be a cogroup in the category of complete -algebras. In this article, we show that R is necessarily the completion of a free -algebra. We also handle the case of cogroups in connected graded algebras over an operad, and the case of groups in connected graded coalgebras over an operad. Received: August 26, 1996 and final version, February 4, 1998     

Up-to-homotopy algebras with strict units

We show that an operad with non-trivial arity zero admits a minimal model in the sense of Sullivan. Hence an up-to-homotopy algebra with a strict unit is just an operad algebra over such a minimal model. We also establish the descent of formality for certain unitary operads. As an application, we give another proof of the formality of the unitary n-little disks operad over the rationals.     

On superalgebras over operads

We show that the class of varieties of multioperator superalgebras, defined by multilinear identities and considered up to rational equivalence, coincides with the class of varieties of superalgebras over an operad. We define the concepts of Grassmann and Clifford envelopes in the most general case and study their properties. We define the concept of module over a superalgebra over an operad and the concept of universal enveloping superalgebra for an algebra over an operad and study their properties.     

Coassociative magmatic bialgebras and the Fine numbers

We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n−2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number F _n−1. In short, the triple of operads (As, Mag, MagFine) is good. The third author work is partially supported by FONDECYT Project 1060224     

La construction bar d'une algèbre comme algèbre de Hopf E-infini

We prove that the bar construction of an E_∞ algebra forms an E_∞ algebra. To be more precise, we provide the bar construction of an algebra over the surjection operad with the structure of a Hopf algebra over the Barratt–Eccles operad. (The surjection operad and the Barratt–Eccles operad are classical E_∞ operads.) To cite this article: B. Fresse, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Symmetrisation of n-operads and compactification of real configuration spaces

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 nonsymmetric 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.     

The operad of framed discs is cyclic

The operad of framed little discs is shown to be equivalent to a cyclic operad. This answers a conjecture of Salvatore, posed at the workshop ‘Knots and Operads in Roma’, at Università di Roma “La Sapienza” in July of 2006, in the affirmative.     

Operads,configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, [`(C)]\bar C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A <sub>∞</sub>-algebras and their homotopy morphisms, (ii) the 2-coloured operad of L <sub>∞</sub>-algebras and their homotopy morphisms, and (iii) the 4-coloured operad of openclosed homotopy algebras and their homotopy morphisms. Two gadgets — a (coloured) operad of Feynman graphs and a de Rham field theory on [`(C)]\bar C — are introduced and used to construct quantized representations of the (fundamental) chain operad of [`(C)]\bar C which are given by Feynman type sums over graphs and depend on choices of propagators.     

Character formulas for the operad of two compatible brackets and for the bi-Hamiltonian operad

We compute the dimensions of the components for the operad of two compatible brackets and for the bi-Hamiltonian operad. We also obtain character formulas for the representations of symmetric groups and SL ₂ in these spaces.     

Minimal models,GT-action and formality of the little disk operad

We give a new proof of formality of the operad of little disks. The proof makes use of an operadic version of a simple formality criterion for commutative differential graded algebras due to Sullivan. We see that formality is a direct consequence of the fact that the Grothendieck–Teichmüller group operates on the chain operad of little disks.     

A Poincaré–Birkhoff–Witt criterion for Koszul operads

The aim of this article is to give a criterion, generalizing the criterion introduced by Priddy for algebras, to prove that an operad is Koszul. We define the notion of Poincaré–Birkhoff–Witt basis in the context of operads. Then we show that an operad having a Poincaré–Birkhoff–Witt basis is Koszul. Besides, we obtain that the Koszul dual operad has also a Poincaré–Birkhoff–Witt basis. We check that the classical examples of Koszul operads (commutative, associative, Lie, Poisson) have a Poincaré–Birkhoff–Witt basis. We also prove by our methods that new operads are Koszul.     

Iterated monoidal categories

We develop a notion of an n-fold monoidal category and show that it corresponds in a precise way to the notion of an n-fold loop space. Specifically, the group completion of the nerve of such a category is an n-fold loop space, and free n-fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher H-space structure of an n-fold loop space. We also show directly that this operad has the same homotopy type as the n-th Smith filtration of the Barratt-Eccles operad and the n-th filtration of Berger's complete graph operad. Moreover, this operad contains an equivalent preoperad which gives rise to Milgram's small model for when n=2 and is very closely related to Milgram's model of for n>2.