Geodesic
An $A_{\infty}$ structure on simplicial complexes
Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 1, pp. 3-37

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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, $n=2$.
Keywords: simplicial complex, topology, discrete exterior form, infinity structure. 
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     author = {V. V. Dolotin and A. Yu. Morozov and Sh. R. Shakirov},
     title = {An $A_{\infty}$ structure on simplicial complexes},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     number = {1},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2008_156_1_a0/}
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V. V. Dolotin; A. Yu. Morozov; Sh. R. Shakirov. An $A_{\infty}$ structure on simplicial complexes. Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 1, pp. 3-37. http://geodesic.mathdoc.fr/item/TMF_2008_156_1_a0/