This question came up when I read german article about operads in topology & homotopy theory. translated to english the statement is:
a $ A_{\infty }$ space is a topological space $Y$ with a homotopy coherent set of maps
$${\displaystyle f_{n}:K_{n}\times Y^{n}\to Y}$$
so that $Y$ is a algebra over the (non symetrical) operad $ K = \{ K_n \}$.
The associahedrons $K_n$ are convex polytopes in which each vertex corresponds to a possible way correctly inserting opening and closing parentheses in a word of $n$ letters and the edges correspond to single application of the associativity rule for $n$ arguments.
Question: what means a homotopy coherent set of maps?
I thought that the edges between vertices of $K_n$ represent the homotopies between different ways to place the parentheses. so what extra data is given by homotopy coherentce of the maps?
This isn’t a very precise statement. What’s actually meant is that $Y$ is given a structure of an algebra over an operad over the $A^\infty$ operad. So the maps $K_n\times Y^n\to Y$ must be compatible with the various maps $K_m\to K_n$ that are part of the operad structure, such as the vertices $K_2\rightrightarrows K_n$ defining the parenthesizations.