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Monoidal Functors, Acyclic Models and Chain Operads

Published online by Cambridge University Press:  20 November 2018

F. Guillén Santos ,
V. Navarro ,
P. Pascual  and
Agustí Roig
F. Guillén Santos
Affiliation:
Departament d’Àlgebra i Geometria, Universitat de Barcelona, 08007 Barcelona, Spain e-mail: fguillen@ub.edu, vicenc.navarro@ub.edu
V. Navarro
Affiliation:
Departament d’Àlgebra i Geometria, Universitat de Barcelona, 08007 Barcelona, Spain e-mail: fguillen@ub.edu, vicenc.navarro@ub.edu
P. Pascual
Affiliation:
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain e-mail: pere.pascual@upc.eduagustin.roig@upc.edu
Agustí Roig
Affiliation:
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain e-mail: pere.pascual@upc.eduagustin.roig@upc.edu
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Abstract

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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.

Keywords

18G8055N1018D50acyclic modelsoperadsmonoidal functorscohomology theories
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 2 , 01 April 2008 , pp. 348 - 378
Copyright
Copyright © Canadian Mathematical Society 2008

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