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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.
[1] Doupovec, M., Kolář, I.: Iteration of fiber product preserving bundle functors. Monatsh. Math. 134 (2001), 39-50. DOI 10.1007/s006050170010 | MR 1872045
[2] Eck, D. J.: Product-preserving functors on smooth manifolds. J. Pure Appl. Algebra 42 (1986), 133-140. DOI 10.1016/0022-4049(86)90076-9 | MR 0857563 | Zbl 0615.57019
[3] Kainz, G., Michor, P. W.: Natural transformations in differential geometry. Czech. Math. J. 37 (1987), 584-607. MR 0913992 | Zbl 0654.58001
[4] Kolář, I.: Weil bundles as generalized jet spaces. Handbook of Global Analysis Elsevier Amsterdam (2008), 625-664 D. Krupka et al. MR 2389643
[5] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer Berlin (1993). MR 1202431
[6] Kolář, I., Mikulski, W. M.: On the fiber product preserving bundle functors. Differ. Geom. Appl. 11 (1999), 105-115. DOI 10.1016/S0926-2245(99)00022-4 | MR 1712139
[7] Kurek, J., Mikulski, W. M.: Fiber product preserving bundle functors of vertical type. Differential Geom. Appl. 35 (2014), 150-155. DOI 10.1016/j.difgeo.2014.04.005 | MR 3254299
[8] Luciano, O. O.: Categories of multiplicative functors and Weil's infinitely near points. Nagoya Math. J. 109 (1988), 69-89. MR 0931952 | Zbl 0661.58007
[9] Weil, A.: Théorie des points proches sur les variétés différentiables. Géométrie différentielle Colloques Internat. Centre Nat. Rech. Sci. 52 Paris French (1953), 111-117. MR 0061455 | Zbl 0053.24903
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