An A∞ 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 ∧. The exterior product is nonassociative but satisfies a more general relation, the so-called A∞ structure. This structure includes an infinite set of operations constrained by the nilpotency relation (d + ∧ + m + …)n = 0 of the second degree, n = 2.