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Lets work in the compactly generated topological spaces category. Let K be the Stasheff's operad (which algebras are $A_\infty$-spaces with strict unit). Please, I would like to understand how K can be given as the relative W-construction $W(I^* \to As)$, where $I^*$ is the operad of pointed spaces and As is the operad of unital monoids.

More precisely, I would like to know what are the specific identifications involving trees in $W(I^* \to As)$.

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    $\begingroup$ Could you describe or at least give a reference for the relative W-construction? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 23, 2017 at 9:20
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    $\begingroup$ The generalized W construction (and its relative version) is given by Berger and Moerdjik in the paper. "The Boardman-Vogt resolution of operads in monoidal model categories". $\endgroup$
    – Izael do Nascimento
    Commented Mar 23, 2017 at 19:17
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    $\begingroup$ If that's your reference, be aware that there is an error in the relative construction in that paper: mathoverflow.net/questions/44166/… $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Mar 24, 2017 at 2:29
  • $\begingroup$ Thank you. But anyway I do not see how the identifications "kills" stumps (with over the "lollipop") in the particular case above in order get the associahedra via trees construction. $\endgroup$
    – Izael do Nascimento
    Commented Mar 24, 2017 at 3:20
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    $\begingroup$ You are correct. That relative $W$-construction, using Berger--Moerdijk, yields infinite dimensional spaces in each arity and certainly does not give the polytopes $K$. In fact, because there is no composition in the operad $I*$ (except with the identity in ${I*}(1)$), the relative-$W$ construction coincides with the ordinary $W$ construction for $As$, which certainly does not give the associahedra because of the unit. $\endgroup$
    – Gabriel C. Drummond-Cole
    Commented Mar 24, 2017 at 3:31

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