How to discretize the differential forms on the interval

TL;DR

This paper constructs explicit quasi-isomorphisms between algebraic structures related to differential forms on the interval, focusing on a natural discretization via $C_$-quasi-isomorphisms and connecting to the Magnus expansion.

Contribution

It introduces a natural discretization $C_$ quasi-isomorphism from differential forms to Whitney forms and provides explicit formulas related to the Magnus expansion.

Findings

Established a uniqueness result for the discretization morphism.

Derived explicit formulas for the $C_$ quasi-isomorphism.

Connected the discretization to classical Magnus expansion formulas.

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 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 $\varphi$ 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 formulas for the Magnus expansion due to Mielnik and Pleba\'nski.