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Natural transformations in differential geometry. (English) Zbl 0654.58001

It is proved that any product-preserving functor from the category of smooth manifolds to itself (satisfying some supplementary conditions) is given by prolongation by a “Weil algebra”, as in A. Weil [Théorie des points proches sur les variétés différentiables, Colloq. Int. Cent. Nat. Rech. Sci. 52, 111-117 (1953; Zbl 0053.249). A similar result was found independently by D. Eck [Product preserving functors on smooth manifolds, J. Pure Appl. Algebra 42, 133- 140 (1986; Zbl 0606.58006)]. The authors utilize this algebraic viewpoint to make some alternative calculations on relations between Lie brackets, covariant differentiation etc. for vector fields. They do not put the theory in a categorical context, where the Weil algebras represent infinitesimal manifolds in their own right, as in, say, E. Dubuc \([C^{\infty}\)-schemes, Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003)].
Reviewer: A.Kock

MSC:

58A05 Differentiable manifolds, foundations
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
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References:

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