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Article

Bandiera, Ruggero ; Schätz, Florian
How to discretize the differential forms on the interval. (English). Higher Structures, vol. 1 (2017), issue 1, pp. 56-86
MSC: 18G55, 55P62 | MR 3912051 | Zbl 1410.18018 | DOI: 10.21136/HS.2017.03
Keywords:
Homotopical algebra; rational homotopy theory; Eulerian idempotent; Magnus expansion; iterated integrals
Summary:
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 $\phi$ 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.
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