Nonstandard diffeology and generalized functions

Abstract

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.