It's well known that the mapping space of two finite dimensional
manifolds can be given the structure of an infinite dimensional manifold
modelled on Frechet spaces (provided the source is compact). However, it is
not that the charts on the original manifolds give the charts on the mapping
space: it is a little bit more complicated than that. These complications
become important when one extends this construction, either to spaces more
general than manifolds or to properties other than being locally linear.
In this talk, I shall show how to describe the type of property needed to
transport local properties of a space to local properties of its mapping
space. As an application, we shall show that applying the mapping
construction to a regular map is again regular.