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Geometrical properties of prolongation functors. (English) Zbl 0582.58002
The prolongation functors generalize the lifting functors or natural bundles in the sense of Nijenhuis. Let \({\mathcal M}\) be the category of all \(C^{\infty}\)-manifolds and \(C^{\infty}\)-maps. Let \({\mathcal C}\) be a subcategory of \({\mathcal M}\) such that every inclusion \(U\hookrightarrow M\), where \(M\in\) ob \({\mathcal M}\) and U is any open set of M, belongs to Hom \({\mathcal C}\). If \({\mathcal F}{\mathcal M}\) is the category of all fibre manifolds and all fibre manifold morphisms, then a prolongation functor on \({\mathcal C}\) is a covariant functor \(F: {\mathcal C}\to {\mathcal F}{\mathcal M}\) satisfying the prolongation condition, a regularity property (that, for lifting functors is a consequence of the prolongation condition) and a localization condition. The author proves some geometrical properties of the prolongation functors, for instance, the equipotency between the set of r-th order prolongation functors and a set of actions of a certain r- jet category. Some known results are pointed out.
Reviewer: I.D.Albu

58A20 Jets in global analysis
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