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On the functorial prolongations of principal bundles. (English) Zbl 1150.58002
It is very well known that if \(Y\to M\) is a fibered bundle associated with a principal bundle \(P\to M\) then \(J^rY\) is the bundle associated with the principal bundle \(W^rP=P^rM\times _M J^rP\), \(P^rM\) being the principal bundle of \(r\)-th order frames. In the paper the authors replace the functor \(J^r\) by an arbitrary fiber product preserving functor \(F\) on the category \({\mathcal {FM}}_m\) of fibered manifolds with \(m\)-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. By using Weil algebras they construct the principal bundle \(W^FP=P^rM\times _M FP\) and prove that \(FY\) is the bundle associated with \(W^FP\).
MSC:
58A20 Jets in global analysis
58A32 Natural bundles
58H05 Pseudogroups and differentiable groupoids
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