Homological algebra for diffeological vector spaces

Abstract

Diffeological spaces are natural generalizations of smooth manifolds, introduced by J.M. Souriau and his mathematical group in the 1980's. As vector spaces, diffeological vector spaces appear canonically from geometry and analysis, and they also contain smooth information. In this paper, we first explore the basic algebraic and categorical constructions on diffeological vector spaces. Then we observe that not every short exact sequence of diffeological vector spaces splits. Motivated by this, we develop the natural analogues of basic tools of classical homological algebra by identifying a good class of projective objects in the category of diffeological vector spaces, together with some applications in analysis. Finally, we prove that there is a cofibrantly generated model structure on the category of diffeological chain complexes.