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Monoidal Functors, Acyclic Models and Chain Operads

  Published:2008-04-01
 Printed: Apr 2008
  • F. GuillĂ©n Santos
  • V. Navarro
  • P. Pascual
  • Agust{\'\i} Roig
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Abstract

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.
MSC Classifications: 18G80, 55N10, 18D50 show english descriptions unknown classification 18G80
Singular theory
Operads [See also 55P48]
18G80 - unknown classification 18G80
55N10 - Singular theory
18D50 - Operads [See also 55P48]
 

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