The Pre-History Of Operads

Mathematical Surveys and Monographs, 2007

Commentarii Mathematici Helvetici, 2003

Advances in Mathematics, 2008

arXiv: Rings and Algebras, 2017

We study different algebraic structures associated to an operad and their relations: to any operad $\mathbf{P}$ is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra; all of them can be explicitely describedwith the help of the operadic composition. non-commutative versions are also given. We denote by $\mathbf{b\_\infty}$ the operad of $\mathbf{b\_\infty}$ algebras, describing all Hopf algebra structures on a symmetric coalgebra.If there exists an operad morphism from $\mathbf{b\_\infty}$ to $\mathbf{P}$, a pair $(A,B)$ of cointeracting bialgebras is also constructed, that it to say:$B$ is a bialgebra, and $A$ is a graded Hopf algebra in the category of $B$-comodules. Most examples of such pairs (on oriented graphs, posets$\ldots$) known in the literature are shown to be obtained from an operad; colored versions of these examples andother ones, based on Feynman graphs, are introduced and compared.

Contemporary Mathematics, 1997

2003

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

2003

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

2001

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.

Journal of the London Mathematical Society

M. Carr and S. Devadoss introduced in [7] the notion of tubing on a finite simple graph Γ, in the context of configuration spaces on the Hilbert plane. To any finite simple graph Γ they associated a finite partially ordered set, whose elements are the tubings of Γ and whose geometric realization is a convex polytope KΓ, the graph-associahedron. For the complete graphs they recovered permutahedra, for linear graphs they got Stasheff's associahedra, while for simple graph they obtained the standard simplexes. The goal of the present work is to give an algebraic description of graph associahedra. We introduce a substitution operation on tubings, which allows us to describe the set of faces of graph-associahedra as a free object, spanned by the set of all connected simple graphs, under operations given via connected subgraphs. The boundary maps of graphassociahedra defines natural derivations in this context. Along the way, we introduce a topological interpretation of the graph tubings and our new operations. In the last section, we show that substitution of tubings may be understood in the context of M. Batanin and M. Markl's operadic categories.