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. $C_{\mathrm{\infty }}$, algebra structure. Our main interest lies in a natural `discretization’ $C_{\mathrm{\infty }}$ quasi-isomorphism $\phi$ from differential forms to Whitney forms. We establish a uniqueness result that implies that $\phi$ coincides with the morphism from homotopy transfer, and obtain several explicit formulas for $\phi$, all of which are related to the Magnus expansion. In particular, we recover combinatorial formulas for the Magnus expansion due to Mielnik and Pleba'nski.