Combinatorial operad actions on cochains

Abstract

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.Comment: 46 pages. Final versio

Dyer–Lashof–Cohen operations in Hochschild cohomology

“…This statement is called "Deligne's conjecture". Over ‫ޚ‬ this result is due to J E McClure and J H Smith [18; 19], and to M Kontsevich and Ya Soibelman [15], see also the proof of C Berger and B Fresse [1], which appeared later. In characteristic zero there are several proofs; see for example M Kontsevich [14], D E Tamarkin [23; 24] and A A Voronov [28].…”

Multivariable cochain operations and little 𝑛-cubes

“…The results of Section 2 have been discovered independently (but with a different choice of signs) in the joint work of Berger and Fresse [3]. After learning Theorem 3.5 from us, they have found a different proof of it using their methods.…”

Differential graded categories

“…The Hochschild cochain complex of a dg category carries a natural homotopy action of the little squares operad. This is the positive answer to a question by P. Deligne [29] which has been obtained, for example, in [105] [92] [13] . .…”