Related papers: Symmetric homotopy theory for operads
We extend the rational homotopy theory of operads developed by B. Fresse to several types of modules over operads.
This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant…
We generalize Berger and Moerdijk's results on axiomatic homotopy theory for operads to the setting of enriched symmetric monoidal model categories, and show how this theory applies to orthogonal spectra. In particular, we provide a…
The present article exploits the fact that permutads (aka shuffle algebras) are algebras over a terminal operad in a certain operadic category Per. In the first, classical part we formulate and prove a claim envisaged by Loday and Ronco…
We explain how the simplicial higher-order unstable homotopy operations defined in [BBS2] may be composed and inserted one in another, thus forming a coherent if complicated algebraic structure.
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics,…
We prove that the category of algebras over a cofibrant operad admits a closed model category structure. This leads to the notion of "virtual operad algebra" - the algebra over a cofibrant resolution of the given operad. In particular,…
The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation…
We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under…
The aim of this note is to give a detailed account of how symmetric operads can be constructed from planar (non-symmetric) operads, and to carefully spell out the algebraic interplay between these two notions. It is a companion note to the…
In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of bialgebras with diagonal symmetries, like double Lie algebras (DLie).…
We present a unified framework for categorical systems theory which packages a collection of open systems, their interactions, and their maps into a symmetric monoidal loose right module of systems over a symmetric monoidal double category…
We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty…
Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using the language of operads. The main goal of this paper is to…
In an application of the notion of twisting structures introduced by Hess and Lack, we define twisted composition products of symmetric sequences of chain complexes that are degreewise projective and finitely generated. Let Q be a cooperad…
The overall aim of this paper is to define a structure of graph operads, thus generalizing the celebrated pre-Lie operad on rooted trees. More precisely, we define two operads on multigraphs, and exhibit a non trivial link between them and…
A new hierarchy of combinatorial operads is introduced, involving families of regular polygons with configurations of arcs, called decorated cliques. This hierarchy contains, among others, operads on noncrossing configurations, Motzkin…
An operad (this paper deals with non-symmetric operads)may be conceived as a partial algebra with a family of insertion operations, Gerstenhaber's circle-i products, which satisfy two kinds of associativity, one of them involving…
The recently proposed differential homotopy approach to the analysis of nonlinear higher spin theory is developed. The Ansatz is extended to the form applicable in the second order of the perturbation theory and general star-multiplication…
The Andr\'e-Quillen cohomology of an algebra with coefficients in a module is defined by deriving a functor based on K\"ahler differential forms. It can be computed using a cofibrant resolution of the algebra in a model category structure…