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


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
Full Text: DOI


[1] van Est, ]Indag. Math 20 pp 399– (1958) · doi:10.1016/S1385-7258(58)50055-9
[2] DOI: 10.2307/2374046 · Zbl 0483.58003 · doi:10.2307/2374046
[3] DOI: 10.1090/S0002-9904-1977-14320-6 · Zbl 0389.58001 · doi:10.1090/S0002-9904-1977-14320-6
[4] Bott, Differential Forms in Algebraic Topology (1982) · Zbl 0496.55001 · doi:10.1007/978-1-4757-3951-0
[5] DOI: 10.1007/BF02564296 · Zbl 0047.16702 · doi:10.1007/BF02564296
[6] Kock, Synthetic Differential Geometry (1981)
[7] Qu?, The L. E. J. Brouwer Centenary Symposium 110 pp 377– (1982) · doi:10.1016/S0049-237X(09)70138-7
[8] Moerdijk, Proceedings of the Workshop on Category Theoretic Methods in Geometry, Aarhus 1983
[9] Lawvere, Cahiers Topologie G?om. Differentielle 21 pp 377– (1980)
[10] Moerdijk, J. Pure Appl. Algebra
[11] DOI: 10.1016/0001-8708(78)90035-X · Zbl 0409.03040 · doi:10.1016/0001-8708(78)90035-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.