Reference for Stasheff's Associahedra Operad

Question

I am currently reading up on operads and am more than confused about the Stasheff operad. It is completely unplausible that I am the first student, who feels this way, so I hope to find a reference explaining to me the following questions in detail.

1. The $n$-th Stasheff Polytope $K_n$ (aka associahedron) is

   • "the convex polytope with one vertex for each way of inserting parantheses into $n$ letters"
   • or "the convex polytope with one vertex for each regular triangulation of a regular $n$-gon $C_n \subseteq \mathbb{R}^2$"
   • or "the convex polytope with one vertex for each binary rooted trees with $n$ leafs".

   Intuitively clear, but not formal. In line with the second reformulation, the construction of the space $K_n$ as secondary polytope makes the most sense to me, since it is easy to verify that a (continuous) section of the canonical morphism $\Delta^{n-1}\rightarrow C_n, e_i \mapsto v_i$ amounts to a triangulation of $C_n$ (more details on this construction here). Assuming a convenient category of spaces in the background, there already is a space of such sections. So how do we motivate taking the barycenters of sections (a posteriori this somehow gives us a convex subset of $\mathbb R^n$, but a priory it is not clear to me, why this is the canonical thing to do)?

2. Why is it clear from this description that $K_n$ always has dimension $n-2$?

3. Most sources I could find just state that the family $(K_n)_n$ of associahedra assemble into a non-symmetric operad. If I am not mistaken, one should specify a bit more than the objects to define an operad. The most detailed description I could find was MSS - Operads in Algebra, Topology and Physics section 1.6, where they claim without proof that

   • the $d$-dimensional cells of $K_n$ are of the form $\prod_{i=1}^k K_{n_i}$, where $d=n_1 + ... + n_k -2k$
   • the facets (i.e. cells of codimension 1, i.e. cells of dimension $n-3$) are of the form $K_r \times K_s$ with $r+s=n+1$

   Again, why is that clear? How do we obtain these facet-embeddings from above description? The operad structure then is given by defining the circle products $\circ_i: K_r \times K_s \subseteq K_{r+s-1}$, which somehow suffices to define the whole operad structure. (I still have to read up on that last part. It seems plausible that this suffices, the only thing worrying me, is that I read somewhere that you cannot always define an operad in this way...)

4. If I am not mistaken, most definitions (wikipedia, nlab) of an operad involve a unit, i.e. contain a choice of element $1 \in \mathcal{O}(1)$ satisfying unitality conditions. I have the wild guess that also the definition of an operad as a monoid involves a unit. Problem is: some sources say that the Stasheff operad is non-unital i.e. doesn't have a unit. (I mean, fair, if one doesn't specify any structure whatsoever, in particular no unit is given, making it canonically non-unital by omission...). This leads to the question, if I define operads to always be unital, is the Stasheff operad even an operad? Or can I at least make it unital?

Remarks

• I don't really care, whether the $K_n$ is a convex polytope or not, for my purposes it would suffice to have an explicit CW-complex of the right dimension. I've heard that you can describe $K_n$ as the space of weighted planar embedded rooted trees with $n$ leafs, where you weight the inner edges by values in the range $[0,1]$. I could not find it in the literature and am unsure, how the topology is defined explicitly. It would clear up points 2. and 3. however, by counting inner edges and grafting trees onto each other.

• The name Stasheff operad is probably non-standard, but to be honest I don't know a better name. This link is related to my question, but doesn't answer it.