We introduce the concept of modified vertical Weil functors on the category \mathcal{F}\mathcal{M}_m of fibred manifolds with m-dimensional bases and their fibred maps with embeddings as base maps. Then we describe all fiber product preserving bundle functors on \mathcal{F}\mathcal{M}_m in terms of modified vertical Weil functors. The construction of modified vertical Weil functors is an (almost direct) generalization of the usual vertical Weil functor. Namely, in the construction of the usual vertical Weil functors, we replace the usual Weil functors T^{a} corresponding to Weil algebras A by the so called modified Weil functors T^{a} corresponding to Weil algebra bundle functors A on the category \mathcal{F}\mathcal{M}_m of m-dimensional manifolds and their embeddings., Włodzimierz M. Mikulski., and Obsahuje seznam literatury
We present a generalization of the concept of semiholonomic jets within the framework of higher order prolongations of a fibred manifold. In this respect, a compilation of our 2-fibred manifold approach with the methods of natural operators theory is used.
Let $\mathcal {P}\mathcal {B}_m$ be the category of all principal fibred bundles with $m$-dimensional bases and their principal bundle homomorphisms covering embeddings. We introduce the concept of the so called $(r,m)$-systems and describe all gauge bundle functors on $\mathcal {P}\mathcal {B}_m$ of order $r$ by means of the $(r,m)$-systems. Next we present several interesting examples of fiber product preserving gauge bundle functors on $\mathcal {P}\mathcal {B}_m$ of order $r$. Finally, we introduce the concept of product preserving $(r,m)$-systems and describe all fiber product preserving gauge bundle functors on $\mathcal {P}\mathcal {B}_m$ of order $r$ by means of the product preserving $(r,m)$-systems.
For every bundle functor we introduce the concept of subordinated functor. Then we describe subordinated functors for fiber product preserving functors defined on the category of fibered manifolds with $m$-dimensional bases and fibered manifold morphisms with local diffeomorphisms as base maps. In this case we also introduce the concept of the underlying functor. We show that there is an affine structure on fiber product preserving functors.