related by the Dold-Kan correspondence
symmetric monoidal (∞,1)-category of spectra
when presenting by the model structure on operads for topological operads, forming the homotopy-algebras over any operad means forming the ordinary algebras over an operad for any of its cofibrant resolutions. Therefore one say: an -operad is (any) cofibrant resolution of Comm in the standard model structure on operads over the model structure on topological spaces.
For every -operad , all the spaces are contractible.
P \to Comm
and for each this is by assumption a weak homotopy equivalence
P_n \to Comm_n = *
of topological spaces.
The only extra condition on an operad with contractible operation spaces to be is that it is in addition cofibrant . This imposes the condition that the action of the symmetric group in each degree is free .
The little k-cubes operad for is .