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

TL;DR

This paper extends the theory of microlinearity and Weil prolongation for Frolicher spaces to develop a general framework for fiber bundles, demonstrating that Weil functors produce fiber bundles and that relevant categories are cartesian closed.

Contribution

It relativizes existing microlinearity theory to fiber bundles for Frolicher spaces, showing Weil functors induce fiber bundles and establishing cartesian closed categories of bundles.

Findings

Weil functors naturally produce fiber bundles.

Categories of fiber and vector bundles over a fixed Frolicher space are cartesian closed.

The tangent bundle functor yields a natural vector bundle.

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.