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$A_\infty$ operad can be described both in terms of Stasheff polytopes and configuration spaces.$A_n$ operad can be described as subspace of Stasheff operad described using Stasheff polytope. Is there a model for $A_n$-operad as a configuration spaces?

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At the risk of saying something stupid I'm promoting my comment to an answer. In terms of the Stasheff polytopes the $A_n$ operad sits inside the $A_\infty$ operad as the union of all faces of dimension $\leq n-2$. Another way of saying this is that the Stasheff polytopes are stratified spaces with strata indexed by rooted trees. The $A_n$ operad is defined as the union of all strata where each vertex of the tree has valence at most $n$.

Now another model of $A_\infty$ operad is given by the sequence of Fulton-MacPherson compactifications of the configuration spaces of points on the interval. The Fulton-MacPherson compactification of any space is also naturally a stratified space, with strata indexed by rooted trees, and the same prescription should work to get a configuration space model of $A_n$ operad.

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  • $\begingroup$ Can you provide a reference for "Fulton-MacPherson compactification" and its naturally "stratefied spaces"? That will be really helpful. $\endgroup$
    – Prasit
    Commented Sep 1, 2013 at 3:49
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    $\begingroup$ I think there are many references for this but I am not sure if I know a good one. I learned this by first reading Fulton and MacPherson's beautiful paper on their construction for complex varieties, which the construction for real manifolds is an imitation of. But maybe you don't want to read a long paper in algebraic geometry. The original reference for the operads is the preprint of Getzler and Jones. It is also explained in some detail in the paper of Lambrechts and Volic on formality of the $n$-disks. $\endgroup$
    – Dan Petersen
    Commented Sep 1, 2013 at 11:27
  • $\begingroup$ I found a quick description of the Fulton-MacPherson compactification in Dror Bar-Natan's notes. $\endgroup$
    – Noah Snyder
    Commented Sep 5, 2013 at 18:33

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