By exploiting the idea of Colombeau 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, asymptotic functions on a Euclidean open set include Schwartz distributions and form a smooth differential algebra over Robinson's field of asymptotic numbers. To illustrate the use of our method, we prove that smooth relative cell complexes enjoy homotopy extension property with respect to asymptotic maps and homotopies. (read more)