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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

Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated

These notes are based on lectures given at the Workshop on Structured ring spectra and

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.

The solution of Deligne’s conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild cochain complex C • (A) of a regular commutative algebra A over a field K of characteristic zero and the Gerstenhaber algebra of multiderivations of A. Unlike the original approach of the second author based on the computation of obstructions our method allows us to avoid the bulky Gelfand-Fuchs trick and prove the formality of the homotopy Gerstenhaber algebra structure on the sheaf of polydifferential operators on a smooth algebraic variety, a complex manifold, and a smooth real manifold.

Abstract. We give a new direct proof of Deligne’s conjecture on the Hochschild cohomology. For this we use the cellular chain operad of normalized spineless cacti as a model for the chains of the little discs operad. Previously, we have shown that the operad of spineless cacti is homotopy equivalent to the little discs operad. Moreover, we also showed that the quasi–operad of normalized spineless cacti is homotopy equivalent to the spineless cacti operad. Now, we give a cell decomposition for the normalized spineless cacti, whose cellular chains form an operad and by our previous results a chain model for the little discs operad. The cells are indexed by bipartite black and white trees which can directly be interpreted as operations on the Hochschild cochains of an associative algebra, yielding a positive answer to Deligne’s conjecture. Furthermore, we show that the symmetric combinations of top–dimensional cells, are isomorphic to the graded pre–Lie operad. Lastly, we define the Hopf algebra of an operad which affords a direct sum. For the pre–Lie suboperad of shifted symmetric top–dimensional chains the symmetric group coinvariants of this Hopf algebra are the renormalization Hopf algebra of Connes and Kreimer.

A conjecture of Deligne stated that the Hochschild cohomology complex of an associative algebra has a natural structure of a 2-algebra, i.e. an algebra over the chain complex version of the 2-cube operad. This indicated a remarkable connection between the deformation theory of associative algebras, and the geometry of