Abstract: I define a particular concept of "semi-weak" omega (infinity) category that I will call "Stasheff category", and attempt to show a close relation between the face structure Stasheff associahedra and the structure of operations on a free Stasheff category. Although the conjecture I have is plausible and is verified in all cases I looked at, I still lack, unfortunately, a complete proof. I will try to relate my ideas with things found in the extensive literature on the Stasheff polyhedra.