Relative Microlinearity -Towards the general theory of fiber bundles for Frolicher spaces

Abstract: In our previous papers [Far East Journal of Mathematical Sciences, 35 (2009), 211-223] and [International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have developed the theory of Weil prolongation, Weil exponentiability and microlinearity for Frolicher spaces. In this paper we will relativize it so as to obtain the theory of fiber bundles for Fr\"olicher spaces. It is shown that any Weil functor naturally gives rise to a fiber bundle. We will see that the category of fiber bundles over a fixed Fr\"olicher space M and their smooth mappings over M is cartesian closed. We will see also that the category of vector bundles over M and their smooth linear mappings over M is cartesian closed. It is also shown that the tangent bundle functor naturally yields a vector bundle.