  We introduce a nonstandard extension of the category of diffeological spaces,
and demonstrate its application to the study of generalized functions. Just as
diffeological spaces are defined as concrete sheaves on the site of Euclidean
open sets, our nonstandard diffeological spaces are defined as concrete sheaves
on the site of open subsets of nonstandard Euclidean spaces, i.e. finite
dimensional vector spaces over (the quasi-asymptotic variant of) Robinson's
hyperreal numbers. It is shown that nonstandard diffeological spaces form a
category which is enriched over the category of diffeological spaces, is closed
under small limits and colimits, and is cartesian closed. Furthermore, it can
be shown that the space of nonstandard functions on the extension of a
Euclidean open set is a smooth differential algebra that admits an embedding of
the differential vector space of Schwartz distributions. Since our algebra of
generalized functions comes as a hom-object in a category, it enables not only
the multiplication of distributions but also the composition of them. To
illustrate the usefulness of this property we show that the homotopy extension
property can be established for smooth relative cell complexes by exploiting
extended maps.
