Monoidal functors, acyclic models and chain operads
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We prove that for a topological operad P the operad of oriented cubical chains, Cord ¤ (P), and the operad of singular chains, S¤(P), are weakly equivalent. As a consequence, Cord ¤ (P;Q) is formal if and only if S¤(P;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 di®erent variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets de¯ned by R-simplicial di®erential graded algebras.