Related papers: Minimal models, GT-action and formality of the lit…
Using a variant of the Boardman-Vogt tensor product, we construct an action of the Grothendieck-Teichm\"uller group on the completion of the little n-disks operad $E_n$. This action is used to establish a partial formality theorem for $E_n$…
We develop the details of Kontsevich's proof of the formality of little N-disks operad over the field of real numbers. Formality holds in the category of operads of chain complexes and also in some sense in the category of commutative…
We extend Tamarkin's formality of the little disk operad to the framed little disk operad.
We prove that the chain operad of small squares is formal. This fact clarifies situation with the proof of M. Kontsevich formality theorem in the paper of the author math.QA/9803025, revised Sept 24. The formality of the operad follows…
We construct a canonical chain of formality quasiisomorphisms for the operad of chains on framed little disks and the operad of chains on little disks. The construction is done in terms of logarithmic algebraic geometry and is remarkable…
The little $n$-disks operad is $SO(n)$ and $O(n)$-equivariantly formal over the rationals. Equivalently, the oriented and unoriented framed little disks operads are rationally formal as $\infty$-operads.
We show that Kontsevich's formality of the little disk operad, obtained using graphs, is homotopic to Tamarkin's formality, for a special choice of a Drinfeld associator. The associator is given by parallel transport of the…
We use Galois group actions on \'etale cohomology to prove results of formality for dg-operads and dg-algebras with torsion coefficients. Our theory applies, among other related constructions, to the dg-operad of singular chains on the…
We prove the existence of minimal models a la Sullivan for operads with non-trivial arity zero. So, up to homotopy algebras with strict units are just operad algebras over these minimal models. As an application we give another proof of the…
We prove the existence of minimal models a la Sullivan for operads with nontrivial arity zero. So up-to-homotopy algebras with strict units are just operad algebras over these minimal models. As an application, we give another proof of the…
We prove the existence of Sullivan minimal models of operad algebras, for a quite wide family of operads in the category of complexes of vector spaces over a field of characteristic zero. Our construction is an adaptation of Sullivan's…
We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichm\"{u}ller group. Using a result of Drummond-Cole, we deduce that the…
We study the action of the orthogonal group on the little $n$-disks operads. As an application we provide small models (over the reals) for the framed little $n$-disks operads. It follows in particular that the framed little $n$-disks…
In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general…
This paper is dedicated to the memory of Moshe Flato, and will appear in Lett. Math. Phys. 48 (1) It became clear during last 5-6 years that the algebraic world of associative algebras (abelian categories, triangulated categories, etc) has…
In this paper, we prove that the group of homotopy automorphisms of the profinite completion of the operad of little $2$-disks is isomorphic to the profinite Grothendieck-Teichm\"uller group. In particular, the absolute Galois group of…
It is shown that every algebra over the chain operad of the little disks operad gives naturally rise to a Hertling-Manin's F-manifold, that is a smooth manifold equipped with an integrable graded commutative associative product on the…
We characterise the profinite Grothendieck-Teichm\"uller group $\widehat{\mathsf{GT}}$ as the group of automorphisms of the profinite completion of a cyclic operad of parenthesised ribbon braids. This operad generates a symmetric monoidal…
In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii)…
This paper is a survey on the homotopy theory of $E_n$-operads written for the new handbook of homotopy theory.