Monoidal Functors, Acyclic Models and Chain Operads (original) (raw)
2008, Canadian Journal of Mathematics
We prove that for a topological operad P the operad of oriented cubical chains, C ord * (P), and the operad of singular chains, S * (P), are weakly equivalent. As a consequence, C ord * (P ; Q) is formal if and only if S * (P ; Q) is formal, thus linking together some formality results that 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. 2 F. GUILLÉN, V. NAVARRO, P.PASCUAL, AND A. ROIG 5. Application: comparison of singular and cubical chains 17 5.1. 17 5.2. A monoidal cotriple in Top 17 5.3. 18 6. Symmetric monoidal functors 19 6.1. Acyclic models for symmetric monoidal functors 19 6.2. The Kleisli cotriple 20 6.3. Ordered cubical chains 20 7. Application to operads 22 7.1. Operads 22 7.2. 23 7.3. Formality 24 7.4. Modular operads 25 8. Contravariant functors 25 8.1. The standard construction 26 8.2. Acyclic models for contravariant monoidal functors 26 8.3. Application to singular and cubical cochains 26 9. Application to cohomology theories 27 9.1. Cohomology theories 27 9.2. 30 References 31
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