Definition Diffeological Space [efr-E1SZ]
Definition Diffeological Space [efr-E1SZ]
A smooth space is a sheaf on \mathsf {SmMfd} in the standard topology of open covers. A smooth space X is a diffeological space if, for each M \in \mathsf {SmMfd}, the map X(M) \to \prod _{p \in M} X(\{p\}) is injective.
A diffeological space with underlying set X is called a diffeology on X, and consists of specifying which maps \mathbb {R}^n \to X are smooth. We call these maps smooth plots.
Given a subset X' \subset X, there is an obvious canonical diffeology on X' given by taking the plots to be those functions whose image in X is smooth. We call this the subspace diffeology.
A morphism of diffeological spaces is called a smooth map. It is equivalently a function X \to Y which carries smooth plots to smooth plots.