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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
##### Keywords:
weak principal bundle
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