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



$A_{\infty}$ structure questions
I have put some notes about operads at 


<http://neil-strickland.staff.shef.ac.uk/research/operads.pdf>


They are not finished (in particular, very many references are missing), but sections 13 and 14 are in reasonable shape, and they should answer your questions. They are written in terms of symmetric operads, so my $K(n)$ consists of $n!$ copies of the Stasheff polytope. 


I have also put another note at 


<http://neil-strickland.staff.shef.ac.uk/research/affop.pdf>


This one shows that various things I had hoped to do with the Stasheff operad are probably not possible.


