  By utilizing the idea of Colombeau's generalized function, we introduce a
notion of asymptotic map between arbitrary diffeological spaces. The category
consisting of diffeological spaces and asymptotic maps is enriched over the
category of diffeological spaces, and inherits completeness and cocompleteness.
In particular, the set of asymptotic functions on a Euclidean open set include
Schwartz distributions and form a Colombeau type smooth differential algebra
over Robinson's field of asymptotic numbers. To illustrate the usefulness of
our machinery, we show that homotopy extension property can be established for
smooth relative cell complexes if we exploit asymptotic maps instead of smooth
ones.
