Differential forms as infinitesimal cochains
type: Item
instance of: scholarly article
title: Differential forms as infinitesimal cochains
zbMATH Open document ID: 0972.58001
DOI: 10.1016/S0022-4049(00)00068-2
author: Anders Kock
published in: Journal of Pure and Applied Algebra
publication date: 2001-11-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58A03
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 18F99
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 1538809
zbMATH Keywords: homotopy equivalence
zbMATH Keywords: de Rham complex of differential forms
reviewed by: George F. Nassopoulos
MaRDI profile type: Publication
cites work: Q4871249
cites work: On De Rham's theorem in synthetic differential geometry
cites work: Q3266916
cites work: Q3918651
cites work: Q4342441
cites work: Q4223539
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description: scientific article; zbMATH DE number 1538809