A definition often found is that an $A_\infty$-algebra is an algebra over an $A_\infty$-operad. here, a
- non-symmetric operad $O$ is $A_\infty$ if all $O(n)$ are contractible
- symmetric operad $O$ is $A_\infty$ if all $O(n)$ are homotopy equivalent to $S_n$, regarded as a discrete space.
As a geometric instance (of the former), one usually finds the associahedron. What would be a corresponding (at best, universal) $A_\infty$ operad valued in chain complexes? In other words, what is the chain-complex model for the associahedron?
Motivation: As for $E_\infty$-operads, one usually finds the Barratt-Eccles operad, which is built from bar resolutions of the Symmetric group, as a universal chain complex-valued $E_\infty$-operad. By the way, what would be the corresponding geometric instance?
The answer can be found in the nLab or in Loday, Valette: Algebraic Operads, §9.2.6.