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Weil algebra; Weil functor; vertical Weil functor; Weil algebra bundle functor; modified Weil functor; modified vertical Weil functor; bundle functor; fiber product preserving bundle functor; natural transformation
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 $\mathcal {M}_m$ of $m$-dimensional manifolds and their embeddings.
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