Hello,

I would like explanation or clear source for some things related to $A_{\infty}$-spaces, via Stasheff's polytopes.

I tried not to think about them, because they seem too complicated for me; I thought that the small $1$-cubes operad, and abstract $A_{\infty}$-operads (each $A(n)$ is contractible), would be enough. But still, when I want to derive, at least for myself, at least heuristically, the axioms of $A_{\infty}$-algebra (in the algebraic sense), I see that I would like to understand those polytopes a bit.

There are different descriptions of $K_n$, Stasheffs polytopes. What would be a clear description, which shows all of the following three features: 1) $K_n$ embed into the small $1$-cubes (non-symmetric) operad; 2) This embedding makes $K_n$ a suboperad. 3) The boundary of $K_n$ breaks to different $K_s \times K_t$, and moreover, I can read the orientations from this, i.e. the signs which I will need to put in the dg-version.

Thank you, Sasha