arXiv:2002.11339v1 [math.AT] 26 Feb 2020 GENERALIZED MAPS BETWEEN DIFFEOLOGICAL SPACES KAZUHISA SHIMAKAWA Abstract. 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 such properties as completeness, cocompleteness, and cartesian closed- ness. In particular, asymptotic functions on a Euclidean open set in- clude 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. 1. Introduction The category Diff of diffeological spaces and smooth maps has excellent properties such as completeness, cocompleteness, and cartesian closedness, and provides a framework for combining analysis with geometry and topol- ogy. But there is one barrier to spread the use of diffeology. Compared with continuous maps, smooth maps are far more difficult to handle with and it often occurs that we cannot find an appropriate smooth map that satis- fies the given requirements. Such a situation typically occurs in analysis, e.g. when solving partial differential equations. Distributions (or general- ized functions) make it possible to differentiate functions whose derivatives do not exist in the classical sense (e.g. locally integrable functions). They are widely used in the theory of partial differential equations, where it may be easier to establish the existence of weak solutions than classical ones, or appropriate classical solutions may not exist. They are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions. The aim of this paper is to bring in such generalized notion of maps into the framework of diffeology, and utilize them to investigate diffeological spaces in a more flexible manner. We now state the main result of the paper. A diffeological space is called a smooth differential algebra if it has a structure of a differential algebra such that its sum, product and partial differential operators are smooth. Given an open subset U ∈ R n , we denote by D ′ (U ) the locally convex topological vector space of Schwartz distributions equipped with the weak dual topology. Then the main theorem of the paper can be stated as follows. Date : February 27, 2020. 2010 Mathematics Subject Classification. Primary 54C35 ; Secondary 46T30, 58D15. This work was supported by JSPS KAKENHI Grant Number JP18K03279. 1