Differential forms as infinitesimal cochains
Inspired by the work of \textit{Y. Félix} and \textit{R. Lavendhomme} on de Rham's theorem [J. Pure Appl. Algebra 69, No. 1, 21-31 (1990; Zbl 0735.51015)], the author discusses in the synthetic setting homotopy equivalence between the de Rham complex of differential forms on a formal manifold \(M\) with a cochain complex, dual to a certain simplicial subcomplex of the singular complex of \(M\), consisting of ``infinitesimal simplices''. Its construction hinges on the notion of the \(n\)th infinitesimal neighbourhood of the diagonal of \(M\). Making even use of ideas from \textit{M. Barr's} article [Theory Appl. Categ. 1, No. 1, 1-9 (1995; Zbl 0849.55003)] as well as from his recent work on the subject, the proof is visualized and reduced to the commutativity of a rectangle of cochain maps defined appropriately. The equivalence so obtained preserves the product structure, by taking wedge product of forms to (the existing) cup product of singular cochains.
scientific article; zbMATH DE number 1538809