  We prove that for a topological operad $P$ the operad of oriented cubical
chains, $C^{ord}_\ast(P)$, and the operad of singular chains, $S_\ast(P)$, are
weakly equivalent. As a consequence, $C^{ord}_\ast(P;\mathbb{Q})$ is formal if
and only if $S_\ast(P;\mathbb{Q})$ is formal, thus linking together some
formality results spread in the literature. The proof is based on an acyclic
models theorem for monoidal functors. We give different variants of the acyclic
models theorem and apply the contravariant case to study the cohomology
theories for simplicial sets defined by $R$-simplicial differential graded
algebras.
