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Cofibrant operads and universal \(E_{\infty}\) operads. (English) Zbl 1076.18007

Summary: We introduce various homotopy structures on the category of operads, which shed some light into the homotopy theoretic nature of the barconstruction \({\mathcal {WB}}\) of an operad, the whiskering process for operads and the \(\Sigma\)-freeness condition. Using the lifting property of cofibrant objects, we construct \(E_\infty\) operads \({\mathcal A}\) which are universal: any \(E_\infty\)-structure lifts to an \({\mathcal A}\)-structure, canonically up to homotopy through \({\mathcal A}\)-structures.

MSC:

18D50 Operads (MSC2010)
55P48 Loop space machines and operads in algebraic topology
Full Text: DOI

References:

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