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This paper deals with the homotopy theory of differential graded operads. We endow the Koszul dual category of curved conilpotent cooperads, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen…

This paper explicitely constructs cofree coalgebras over operads in the category of DG-modules. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). It is shown that the existence…

The purpose of this foundational paper is to introduce various notions and constructions in order to develop the homotopy theory for differential graded operads over any ring. The main new idea is to consider the action of the symmetric…

For operads with a map from the curved homotopy Lie operad, we introduce a corresponding curved variant `cTw' of Willwacher's operadic twisting comonad `Tw'. We show that cTw-coalgebra structures on such an operad are in bijection with…

In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large…

We introduce $k$-signaletic operads and their Koszul duals, generalizing the dendriform, diassociative and duplicial operads (which correspond to the $k=1$ case). We show that the Koszul duals of the $k$-signaletic operads act on…

We study the operad structure on the homology of moduli spaces of pointed rooted trees of $d$-dimensional projective spaces, introduced by Chen, Gibney and Krashen a couple of decades ago. We describe this operad by generators and…

In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general…

This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also…

We define a construction on operads which yields a new description of the minimal model. The construction also allows us to define algebraic structures on the homology of chain complexes with homologously trivial operad algebra structures,…

The goal of this paper is to prove a Koszul duality result for E_n-operads in differential graded modules over a ring. The case of an E_1-operad, which is equivalent to the associative operad, is classical. For n>1, the homology of an…

The associative operad is a central structure in operad theory, defined on the linear span of the set of permutations. We build two analogs of the associative operad on the linear span of the set of packed words which turn out to be…

We investigate algebras with one operation. We study when these algebras form a monoidal category and analyze Koszulness and cyclicity of the corresponding operads. We also introduce a new kind of symmetry for operads, the dihedrality,…

We complete a certain diagram (the operadic butterfly) of categories of algebras involving Com, As, and Lie by constructing a type of algebras which have 4 generating operations and 16 relations. The associated operad is self-dual for…

We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial…

We apply the theory of operadic Koszul duality to provide a cofibrant resolution of the colored operad whose algebras are prefactorization algebras on a fixed space M. his allows us to describe a notion of prefactorization algebra up to…

By an odd structure we mean an algebraic structure in the category of graded vector spaces whose structure operations have odd degrees. Particularly important are odd modular operads which appear as Feynman transforms of modular operads…

We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories…

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…

This paper introduces the structure of operated polygraphs as a categorical model for rewriting in operated algebras, generalizing Gr\"obner-Shirshov bases with non-monomial termination orders. We provide a combinatorial description of…