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.

Abstract. Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley,..., Toën and Toën-Vaquié. 1.

Abstract. In this paper we construct a small E ∞ chain operad S which acts naturally on the normalized cochains S ∗ X of a topological space. We also construct, for each n, a suboperad Sn which is quasi-isomorphic to the normalized singular chains of the little n-cubes operad. The case n = 2 leads to a substantial simplification of our earlier proof of Deligne’s Hochschild cohomology conjecture. 1. Introduction. This paper has two goals. The first (see Theorem 2.15 and Remark 2.16(a)) is to construct a small E ∞ chain operad S which acts naturally on the normalized cochains S∗X of a topological space X. This is of interest in view of a theorem of Mandell [15, page 44] which states that if O is any E ∞ chain operad over Fp (the algebraic closure of the field with

Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.

Abstract. Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology constructed by M. Chas and D. Sullivan, V. Godin and R. Cohen. We generalize some of these operations to spaces of maps from a sphere to a compact manifold. 1.

Abstract. In this paper and its follow-up [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L∞-algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.

We proved in a previous article that the bar complex of an E ∞-algebra inherits a natural E ∞-algebra structure. As a consequence, a well-defined iterated bar construction B n (A) can be associated to any algebra over an E ∞-operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E ∞-algebras. We use this effective definition to prove that the n-fold bar construction admits an extension to categories of algebras over En-operads. Then we prove that the n-fold bar complex determines the homology theory associated to the category of algebras over an En-operad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γ-homology with trivial coefficients. 57T30; 55P48, 18G55, 55P35

Abstract. In this paper, we introduce the Adem-Cartan operad related to level algebras and E∞-structures. This operad enables an explicit proof of the Cartan-Adem relations for the Steenrod squares (at the prime p = 2). Moreover, it gives an operadic approach to secondary cohomological operations.

Abstract. 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 an operad. In view toward applications, we check that the constructions of the bar duality work properly for algebras over operads in unbounded differential graded modules over a ring. In a second part, we use the operadic cobar construction to define explicit cyclinder objects in the category of operads. Then we apply this construction to prove that certain homotopy morphisms of algebras over operads are equivalent

Abstract. In this paper, we introduce Adem-Cartan operads and prove that the cohomology of any algebra over such an operad is an unstable level algebra over the extended Steenrod algebra. Moreover we prove that this cohomology is endowed with secondary cohomological operations. MSC (2000): 55-xx; 55S10; 18D50. Keywords: cohomology operations, Adem-Cartan relations, operads, E∞-structures, level algebras.