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De Rham’s theorem in a smooth topos. (English) Zbl 0582.55006

The authors consider cohomological properties of manifolds, but of a much larger category \({\mathcal G}\) of smooth spaces. The authors’ conceptions about \({\mathcal G}\) and more generally about any Grothendieck topos are given in the introduction: ”The rather complex mathematical structure of the topos can be kept away from the language, which is now used to describe what goes on ’inside’ the topos, rather than talk about the topos from an ’external’, classical point of view. Furthermore, this ’constructive’ set-theoretical reasoning within a topos opens the way to regard the topos as a model of a theory which may by inconsistent with classical logic.”
In this context the authors consider objects M, called ”smooth spaces” and define the n-dimensional singular homology R-module of M completely parallel to the classical definitions. They give extensions of De Rham’s theorem to the topos \({\mathcal G}\) and show that this theorem also holds for other cohomologies (Čech cohomology, singular cohomology, etc.). In the last paragraph the authors give different versions of De Rham’s theorem in \({\mathcal G}\). Let us mention De Rham’s theorem with parameters and De Rham’s theorem in \({\mathcal G}\) for continuous homology.
Reviewer: M.Marinov

MSC:

55N10 Singular homology and cohomology theory
58A12 de Rham theory in global analysis
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
18B25 Topoi
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