On the fiber product preserving bundle functors. (English) Zbl 0935.58001

Kolář, Ivan; Mikulski, Wlodzimierz M.

Differ. Geom. Appl. 11, No. 2, 105-115 (1999).

Let \(Mf\) be the category of all manifolds and all smooth maps, \(Mf_m\) be the category of \(m\)-dimensional manifolds and their local diffeomorphisms and \(FM_m\) be the category of fibered manifolds with \(m\)-dimensional bases and fiber preserving maps with the base map in \(Mf_m\).
The authors deduce a complete description of all fiber product preserving bundle functors on \(FM_m\) in terms of Weil algebras. The main result reads that every such functor has finite order and all such functors of order \(r\) are in bijection with triples \((A,H,t)\), where \(A\) is a Weil algebra of order \(r\), \(H\) is a group homomorphism from the \(r\)-th jet group \(G^r_m\) into the group of all algebra automorphisms of \(A\) and \(t\) is a \(G^r_m\)-invariant algebra homomorphism from the jet algebra \(J^r_0(\mathbb{R}^m,\mathbb{R})\) into \(A\). The proof is based on several properties of the bundle functors on the product category \(Mf_m\times Mf\) that are deduced in the first part of the paper.

Reviewer: Ivan Kolář (Brno)

MSC:

58A05	Differentiable manifolds, foundations
58A20	Jets in global analysis

Keywords:

Weil bundle; jet bundle; natural transformation; bundle functors

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