Cubical cochains are equipped with an associative product, dual to the Serre diagonal, lifting the graded ring structure in cohomology. In this work we introduce through explicit combinatorial methods an extension of this product to a full $E_infty$-
structure. We also study the Cartan-Serre map relating the cubical and simplicial singular cochains of spaces, and prove that this classical map is a quasi-isomorphism of $E_infty$-algebras.
Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-$i$ products; a family of coherent homotopies derived from the broken symmetry of Alexander-Whitneys chain approximation to th
e diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of cohomology, having definitions of Steenrod operations that can be effectively computed in specific examples has become a key issue. This article provides such definitions at all primes using the operadic viewpoint of May, and defines multioperations that generalize the cup-$i$ products of Steenrod on the simplicial and cubical cochains of spaces.
