This paper is an introduction to the series of papers [26, 28, 29, 30], in which we develop a combinatorial theory of certain important operads and their actions. The operads we consider are A∞ operads, E∞ operads, the little n-cubes operad and the framed little disks operad. Sections 2, 6 and 9, which can be read independently, are an introduction to the theory of operads. The reader is also r...

Abstract. Coherent unit actions on a binary, quadratic operad were introduced by Loday and were shown by him to give Hopf algebra structures on the free algebras when the operad is also regular with a splitting of associativity. Working with such operads, we characterize coherent unit actions in terms of linear equations of the generators of the operads. We then use these equations to give all ...

Two notions, generic morphisms and parametric representations, useful for the analysis of endofunctors arising in enumerative combinatorics, higher dimensional category theory, and logic, are defined and examined. Applications to the Batanin approach to higher category theory, Joyal species and operads are provided.

We study the divided power structures over a product of operads with distributive law. give systematic method to characterise algebras such from algebra coming each factor operads. operadic derivation, as well $p$-level in characteristic $p$, and Poisson 3.

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. Mathematics Subject Classification (2000). Primary 18D50; Secondary 18G55, 55U35.

1 Foundations 5 1.1 The E[k]-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 The Additivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Iterated Loop Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Coherence of the Little Cubes Operads . . . . . . ....

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.

This text is an introduction to operads from a combinatorial point of view. We give the definition, several examples of operads and the construction of a group from each operad.