An $A_{\infty}$ structure on simplicial complexes

We consider a discrete (finite-difference) analogue of differential forms defined on simplicial complexes, in particular, on triangulations of smooth manifolds. Various operations are explicitly defined on these forms including the exterior differential $d$ and the exterior product $\wedge$. The exterior product is nonassociative but satisfies a more general relation, the so-called $A_{\infty}$ structure. This structure includes an infinite set of operations constrained by the nilpotency relation $(d+\wedge+m+\dotsb)^n=0$ of the second degree, $n=2$.