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How to discretize the differential forms on the interval

We provide explicit quasi-isomorphisms between the following three algebraic structures associated to the unit interval: i) the commutative dg algebra of differential forms, ii) the non-commutative dg algebra of simplicial cochains and iii) the Whitney forms, equipped with a homotopy commutative and homotopy associative, i.e. $C_\infty$, algebra structure. Our main interest lies in a natural `discretization' $C_\infty$ quasi-isomorphism $φ$ from differential forms to Whitney forms. We establish a uniqueness result that implies that $φ$ coincides with the morphism from homotopy transfer, and obtain several explicit formulas for $φ$, all of which are related to the Magnus expansion. In particular, we recover combinatorial formulas for the Magnus expansion due to Mielnik and Plebański.