What is "diffeology"

In mathematics, a diffeology on a set declares what the smooth parametrizations in the set are. In some sense a diffeology generalizes the concept of smooth charts in a differentiable manifold.

The concept was first introduced by Jean-Marie Souriau in the 1980s and developed first by his students Paul Donato (homogeneous spaces and coverings) and Patrick Iglesias (diffeological fiber bundles, higher homotopy etc.), later by other people. A related idea was introduced by Kuo-Tsaï Chen (陳國才, Chen Guocai) in the 1970s, using convex sets instead of open sets for the domains of the plots.