Abstract. We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories of pointed plots are (weakly) filtered. We extend the exact sequence one step further in the case of a diffeological bundle with filtered total space and base space. We also show that the tangent bundle T H X defined by Hector is a diffeological vector space over X when X is filtered or when X is a homogeneous space, and therefore agrees with the dvs tangent bundle introduced by the authors in a previous paper.

Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the D-topology. However, the D-topology has not yet been studied seriously in the existing literature. In this paper, we develop the basic theory of the D-topology for diffeological spaces. We explain that the topological spaces that arise as the D-topology of a diffeological space are exactly the ∆-generated spaces and give results and examples which help to determine when a space is ∆-generated. Our most substantial results show how the D-topology on the function space C ∞ (M, N ) between smooth manifolds compares to other well-known topologies.

Diffeological spaces are natural generalizations of smooth manifolds, introduced by J.M. Souriau and his mathematical group in the 1980's. Diffeological vector spaces (especially fine diffeological vector spaces) were first used by P. Iglesias-Zemmour to model some infinite dimensional spaces in [I1, I2]. K. Costello and O. Gwilliam developed homological algebra for differentiable diffeological vector spaces in Appendix A of their book [CG]. In this paper, we present homological algebra of general diffeological vector spaces via the projective objects with respect to all linear subductions, together with some applications in analysis.

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 Frölicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frölicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frölicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Frölicher 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. Mathematics Subject Classification

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