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 …

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 …

The purpose of this paper is twofold. First, we review applications of the bar duality of
operads to the construction of explicit cofibrant replacements in categories of algebras over …

In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term
of arity one is not necessarily reduced to an operadic unit, extending results obtained by the …

This paper proves Koszul duality for coloured operads and uses it to introduce strongly
homotopy operads as a suitable homotopy invariant version of operads. It shows that …

In many areas of mathematics some “higher operations” are arising. These havebecome so
important that several research projects refer to such expressions. Higher operationsform …

The goal of the present paper is to compare, in a precise way, two approaches of operads
up to homotopy which appear in the literature. Namely, we construct a functor from the …

We introduce a general definition for coloured cyclic operads over a symmetric monoidal
ground category, which has several appealing features. The forgetful functor from coloured …

We present a definition of homotopy algebra for an operad, and explore its consequences.
The paper should be accessible to topologists, category theorists, and anyone acquainted …

This paper studies the homotopy theory of algebras and homotopy algebras over an operad.
It provides an exhaustive description of their higher homotopical properties using the more …