Reference for Stasheff Operad

I want a reference for Stasheff operad, where operad maps are defined explicitly at the point-set level. I would also like to ask the question that what exactly do one mean by Stasheff operad? Is there a specific pointset model that one thinks or is the description in terms of trees good enough to call that a Stasheff operad?

EDIT: If I choose my own model of Stasheff polytope and my own set of maps which obeys the operad laws and at the cellular level the maps remain the same. Can I say such an operad to be "Stasheff operad"? Or do people mean something more/less specific by the term "Stasheff operad"? Or is it the case that there is no literature where the operad structure is worked out explicitly and the term Stasheff operad is used but never defined explicitly, people just chose there favorite model when they had to work with it.

• "Or is it the case that there is no literature where the operad structure is worked out explicitly and the term Stasheff operad is used but never defined explicitly" -- This really makes no sense to me. Of course there are explicit concrete descriptions, as I indicated in my answer. There is also a universal property description, where $K$ is initial among non-unital topological operads $M$ for which each $M_n$ is equipped with a basepoint and specified contracting homotopy. So if your description has this universal property, then that is good enough. – Todd Trimble Sep 5 '13 at 22:37
• I added some details on the universal property to my answer. – Todd Trimble Sep 6 '13 at 13:53

Why not look at Stasheff's original paper? He does give a point-set model (where $K_{n+2}$ is a compact convex semialgebraic subset of $\mathbb{R}^n$) and describes explicitly the substitution maps $\text{sub}_i: K_m \times K_n \to K_{m+n-1}$ which are collectively tantamount to the operad structure.

• James Dillon Stasheff, Homotopy Associativity of H-Spaces I, Transactions of the American Mathematical Society Vol. 108, No. 2 (Aug., 1963), pp. 275-292.

There are other descriptions which realize the associahedra as linear polytopes (which I find more pleasant); the name of the late J.-L. Loday sticks prominently in my mind in this regard. See here for instance.

Edit: The universal property description of the associahedral operad (or Stasheff operad; I've used both terms), mentioned in my comment under the question, might not be well-known; it might be good to describe intuitively what is going on there. I should add that I have no idea where this observation might appear in the literature, but that I would also have a hard time believing that the observation is originally due to me. :-) See this $n$-Category Café post for some related background.

The statement is that the Stasheff operad is initial among non-permutative non-unital operads $M$ such that each $M_n$, for $n \geq 2$, carries a basepoint and an $I$-module map $I \times M_n \to M_n$ (for the multiplicative monoid $I = [0, 1]$) that contracts $M_n$ to that basepoint. It should be noted that the correct definition of non-unital operad is one that uses operations $sub_i: M_m \times M_n \to M_{m+n-1}$ for $1 \leq i \leq m$, not a less expressive definition which merely involves a structure $M \circ M \to M$ where $\circ$ is the well-known substitution product on graded spaces (see Tom Leinster's book for details on such substitution products, monoids of which being unital operads). In the Stasheff model, we could take the barycenter of each associahedron as its basepoint $p$; the contracting homotopy is given by $x \mapsto t x + (1-t)p$, taking advantage of convexity for the Stasheff model.

In the initial such structure, there is no operad identity, so $M_1$ is empty, and so is $M_0$. For $M_2$, all we know is that it has a basepoint $m_2$ (so $M_2$ is the cone on an empty space). For $M_3$, all we know is that there are points $l = sub_1(m, m)$ (think $(xy)z$) and $r = sub_2(m, m)$ (think $x(yz)$) plus a contracting homotopy to a basepoint $m_3$, so $M_3$ is a cone on the space $\{l, r\}$. This cone is of course an interval. Continuing up the inductive ladder, $M_4$ is obtained by taking the cone of a space obtained by pasting together five spaces corresponding to images $sub_1(M_2 \times M_3)$, $sub_2(M_2 \times M_3)$, $sub_1(M_3 \times M_2)$, $sub_2(M_3 \times M_2)$, $sub_3(M_3 \times M_2)$; the pasting is in accordance with generalized operadic associativity. The pasting gives the boundary of a pentagon; the cone itself is the famous Stasheff pentagon. And so on.

• Please see the edit. Thanks for the replies, I tried to be more precise with the question I have in mind. – Prasit Sep 5 '13 at 22:18
• I was reading through the Stasheff's paper where he produces degeneracy maps (prop 3). I was wondering if that makes Stasheff operad into an unital operad, i.e. $K(0) = *$. – Prasit Sep 22 '13 at 14:31
• What is the similar universal operad in permutative non-unital contractible operads? Dose it have a similar linear model? – Mostafa Mar 15 '15 at 11:36
• @Mostafa If you really mean "contractible" instead of [an] "acyclic" [structure], so that each $M_n$ is in particular connected and there is for example a "homotopy" between the formal operations "$xy$" and "$yx$" in $M_2$, then I think one gets the permuto-associahedron; see for example here: sciencedirect.com/science/article/pii/002240499390049Y (the main theorem exhibits a convex linear polytope). – Todd Trimble Mar 15 '15 at 12:19

There's a point-set model given in terms of a compactification of the configuration space of points in an interval $[0,1]$. This is popular in the "spaces of knots" literature that uses the Goodwillie-Weiss embedding calculus.

Dev's paper is probably not geodesically going towards what you want, but it's a good starting point:

Dev Sinha. Manifold-theoretic compactifications of configuration spaces. Selecta Mathematica (new series) Vol 10, No 3 (2004) 391-428.

Another related paper is Paolo Salvatore's:

Paolo Salvatore. Configuration spaces with summable labels. Cohomological Methods in Homotopy Theory, Prog. in Math. 196 (2001) pp. 375-396.

Where he gives descriptions of monoidal bar constructions in this language.

I have not heard people refer specifically to "The Stasheff operad" very often. Usually people talk about "the Stasheff polytope" or "the associahedron". But my impression is people mean "the Stasheff operad" as the operad whose operations are parametrized by the Stasheff associahedra.

• Please see the edit. Thanks for the replies, I tried to be more precise with the question I had. – Prasit Sep 5 '13 at 22:19