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.     

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.     

The Eckmann-Hilton argument and higher operads

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category.In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann-Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, … , one (n−1)-arrow algebras of A is isomorphic to the category of algebras of Symn(A). Under some mild conditions, we present an explicit formula for Symn(A) which involves taking the colimit over a remarkable categorical symmetric operad.We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series.     

Operads within Monoidal Pseudo Algebras

A general notion of operad is given, which includes: (1) the operads that arose in algebraic topology in the 1970s to characterise loop spaces. (2) the higher operads of Michael Batanin [4] (3) braided and symmetric analogues of Batanin’s operads which are likely to be important in the study of weakly symmetric higher dimensional monoidal categories. The framework of this paper, links together two-dimensional monad theory, operads, and higher dimensional algebra, in a natural way.     

The operad Lie is free

We show that the operad Lie is free as a non-symmetric operad. Then we study the generating series counting the operadic generators. We find a recursive formula for the coefficients of the series and show that the asymptotic density of the operadic generators is 1/e.     

The homotopy theory of bialgebras over pairs of operads

We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second step we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes.     

A Grothendieck module with applications to Poincaré rationality

In this paper, we define a Grothendieck module associated to a Noetherian ring A. This structure is designed to encode relations between A-modules which can be responsible for the relations among Betti numbers and therefore rationality of the Poincaré series. We will define the Grothendieck module, demonstrate that the condition of being torsion in the Grothendieck module implies rationality of the Poincaré series, and provide examples. The paper concludes with an example which demonstrates that the condition of being torsion in the Grothendieck module is strictly stronger than having rational Poincaré series.     

Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a category of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of diassociative algebras, called γ-pluriassociative algebras, so that 1-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with 2γ associative binary operations satisfying some relations. We provide a complete study of the γ-pluriassociative operads, the underlying operads of the category of γ-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in γ-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.     

Operads and varieties of algebras defined by polylinear identities

We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of C-linear multioperator algebras; in particular, of algebras, defined by C-polylinear identities (here C is a commutative operad). We introduce and study symmetric C-linear operads. The main result of this article is as follows: A variety of C-linear multioperator algebras is defined by C-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric C-linear operad.     

Pluriassociative algebras II: The polydendriform operad and related operads

Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform operad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of dendriform algebras, called γ-polydendriform algebras, so that 1-polydendriform algebras are dendriform algebras. For that, we consider the operads obtained as the Koszul duals of the γ-pluriassociative operads introduced by the author in a previous work. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, γ-polydendriform algebras seem adapted structures to split associative operations into 2γ operation so that some partial sums of these operations are associative. We provide a complete study of the γ-polydendriform operads, the underlying operads of the category of γ-polydendriform algebras. We exhibit several presentations by generators and relations, compute their Hilbert series, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer parameter of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.     

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.     

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.     

Algebras with bracket

An algebra with bracket is an associative algebra A equipped with a bilinear operation [−,−] satisfying [a · b, c] = [a, c]·b+a · [b, c]. Our main result claims that the operad corresponding to algebras with bracket is Koszul.     

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).     

Generating sequences and Poincaré series for a finite set of plane divisorial valuations

Let V be a finite set of divisorial valuations centered at a 2-dimensional regular local ring R. In this paper we study its structure by means of the semigroup of values, SV, and the multi-index graded algebra defined by V, grVR. We prove that SV is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in V, the approximation of a reduced plane curve singularity C by families of sets V(k) of divisorial valuations, and the relationship between the value semigroup of C and the semigroups of the sets V(k), allow us to obtain the (finite) minimal generating sequences for C as well as for V.We also analyze the structure of the homogeneous components of grVR. The study of their dimensions allows us to relate the Poincaré series for V and for a general curve C of V. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincaré series of V. Moreover, the Poincaré series of C could be seen as the limit of the series of V(k), k?0.     

André–Quillen cohomology of algebras over an operad

We study the André–Quillen cohomology with coefficients of an algebra over an operad. Using resolutions of algebras coming from the Koszul duality theory, we make this cohomology theory explicit and we give a Lie theoretic interpretation. For which operads is the associated André–Quillen cohomology equal to an Ext-functor? We give several criteria, based on the cotangent complex, to characterize this property. We apply it to homotopy algebras, which gives a new homotopy stable property for algebras over cofibrant operads.     

The Boardman-Vogt resolution of operads in monoidal model categories

We extend the -construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed Σ-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised -construction.     

Operads of hypergraphs

We construct two countable collections of operads of multidimensional cubic matrices. In every operad from one of these collections we select a suboperad such that its elements can be interpreted as some hypergraphs. We prove that all these suboperads are Epi-operads.