Categorical frameworks for generalized functions

We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Fr\"olicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Fr\"olicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Fr\"olicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Fr\"olicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions.Comment: 38 page

Pushforward and projective diffeological vector pseudo-bundles

In this paper, we study a new operation named pushforward on diffeological vector pseudo-bundles, which is left adjoint to the pullback. We show how to pushforward projective diffeological vector pseudo-bundles to get projective diffeological vector spaces, producing many concrete new examples. This brings new objects to diffeology from classical vector bundle theory

A Homotopy Theory for Diffeological Spaces

Smooth manifolds are central objects in mathematics. However, the category of smooth manifolds is not closed under many useful operations. Since the 1970\u27s, mathematicians have been trying to generalize the concept of smooth manifolds. J. Souriau\u27s notion of diffeological spaces is one of them. P. Iglesias-Zemmour and others developed this theory, and used it to simplify and unify several important concepts and constructions in mathematics and physics. We further develop the diffeological space theory from several aspects: categorical, topological and differential geometrical. Our main concern is to build a suitable homotopy theory (also called a model category structure) on the category of diffeological spaces, which encodes the usual smooth homotopy theory of smooth manifolds and the diffeological bundle theory of Iglesias. This is a huge task, and at the moment, we have not yet completely proved the existence theorem. However, in the process, we can see the beauty of the merging of differential geometry and homotopy theory. (More details are explained in the Introduction.) These results should be of some interest to people working in these fields