Monoidal Functors, Acyclic Models and Chain Operads

Santos, F. Guillén ; Navarro, V. ; Pascual, P. ; Roig, Agustí

Canadian journal of mathematics, Tome 60 (2008) no. 2, pp. 348-378

Résumé

We prove that for a topological operad $P$ the operad of oriented cubical singular chains, $C_{*}^{^{\text{ord}}}(P)$ , and the operad of simplicial singular chains, ${{S}_{*}}(P)$ , are weakly equivalent. As a consequence, $C_{*}^{^{\text{ord}}}(P;\,\mathbb{Q})$ is formal if and only if ${{S}_{*}}(P;\,\mathbb{Q})$ 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.

Détail

DOI : 10.4153/CJM-2008-017-7

Mots-clés : 18G80, 55N10, 18D50, acyclic models, operads, monoidal functors, cohomology theories

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