  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.
