Diffeology

Summary

A diffeology on a set declares which maps from open subsets of Euclidean spaces into the set count as smooth; these maps are called plots. A set with a diffeology is a diffeological space. Every smooth manifold is a diffeological space, but so are quotients, subsets and spaces of smooth maps, which are often not manifolds. Diffeological spaces form a cartesian closed category with all limits and colimits, so constructions that leave the world of manifolds stay inside this one.