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Search: MSC category 18D50 ( Operads [See also 55P48] )

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1. CJM 2008 (vol 60 pp. 348)

Santos, F. Guillén; Navarro, V.; Pascual, P.; Roig, Agust{\'\i}
Monoidal Functors, Acyclic Models and Chain Operads
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.

Categories:18G80, 55N10, 18D50

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