### How to discretize the differential forms on the interval

#### Abstract

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 simplicialcochains 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$\varphi$ from differential forms to Whitney forms.We establish a uniqueness result that implies that $\varphi$coincides with the morphism from homotopy transfer,and obtain several explicit formulas for $\varphi$, all of which are related to the Magnus expansion.In particular, we recover combinatorial formulasfor the Magnus expansion due to Mielnik and Pleba\'nski.

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