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Monoidal functors, acyclic models and chain operads
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We prove that for a topological operad $P$ the operad of oriented cubical singular chains, $C^{\ord}_\ast(P)$, and the operad of simplicial singular chains, $S_\ast(P)$, are weakly equivalent. As a consequence, $C^{\ord}_\ast(P\nsemi\mathbb{Q})$ is formal if and only if $S_\ast(P\nsemi\mathbb{Q})$ is formal, thus linking together some formality results which are spread out 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.
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GUILLÉN SANTOS, Francisco, et al. Monoidal functors, acyclic models and chain operads. Canadian Journal of Mathematics-Journal Canadien de Mathematiques. 2008. Vol. 60, num. 348-378. ISSN 0008-414X. [consulted: 9 of October of 2026]. Available at: https://hdl.handle.net/2445/34464