2016/07/13 by Ruggero Bandiera, Bandiera, Ruggero, Schaetz, Florian
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1607.03654
openalex publication_date 2016/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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_∞, algebra structure. Our main interest lies in a natural `discretization' C_∞ 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.