Weilian Prolongations of Actions of Smooth Categories

First of all, we find some further properties of the characterization of fiber product preserving bundle functors on the category of all fibered manifolds in terms of an infinite sequence A of Weil algebras and a double sequenceH of their homomorphisms from [5]. Then we introduce the concept of Weilian prolongation WA HS of a smooth category S over N and of its action D. We deduce that the functor (A,H) transforms D-bundles into WA HD-bundles. In [4] we clarified that every fiber product preserving bundle functor on the category FMm of fibered manifolds with m-dimensional bases and fiber preserving morphisms with local diffeomorphisms as base maps is of finite order and can be identified with a triple (Am, Hm, tm), where Am is a Weil algebra, Hm : Gm → Aut Am is a group homomorphism of the r-th jet group in dimension m into the group of all algebra automorphisms of Am and tm : Dm → Am is an equivariant algebra homomorphism, Dm = J 0 (R,R). Our next result from [5] can be formulated as follows. Write (1) A = (A1, . . . , Am, . . . ) for an infinite sequence of Weil algebras, (2) Hom A = ( Hom (Am, An) ) for the double sequence of the algebra homomorphisms and (3) L = (Lm,n) , Lm,n = J 0 (R,R)0 for the skeleton of the category of r-jets. Then the fiber product preserving bundle functors F on the category FM of all fibered manifold morphisms of the base order r are in bijection with the pairs (A,H) of a sequence (1) and of a functor (4) H : L → Hom A , Hm,n : Lm,n → Hom (Am, An) . In the first two sections of the present paper, we deduce certain new results concerning F and an arbitrary fiber product preserving bundle functor on FMm, that are to be used in the sequel. In Section 3 we consider a smooth category S over 2000 Mathematics Subject Classification: Primary: 58A20; Secondary: 58A32.