Monoidal functors, acyclic models and chain operads

Details Monoidal functors, acyclic models and chain operads Monoidal functors, acyclic models and chain operads

2005-06-30

arxiv.org/abs/math/0506624v2

Mathematics

Algebraic Topology

29 pages; introduction improved, two changes in the proofs of theorems 5.3.1 and 6.3.2

Scientific paper

We prove that for a topological operad $P$ the operad of oriented cubical chains, $C^{ord}_\ast(P)$, and the operad of singular chains, $S_\ast(P)$, are weakly equivalent. As a consequence, $C^{ord}_\ast(P;\mathbb{Q})$ is formal if and only if $S_\ast(P;\mathbb{Q})$ is formal, thus linking together some formality results spread 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.

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