Monoidal functors, acyclic models and chain operads

2005/06/30 by F. Guillen Santos, V. Navarro, Santos, F. Guillen +5
Mathematics · #18G60 #55N40 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #math.AT #math.CT #msc:18G60 #msc:55N40

paper · pdf · doi:10.48550/arxiv.math/0506624

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

arxiv created 2005/10/03 · arxiv updated 2009/12/01

Abstract

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 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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