The notion of diffeological space is due to Souriau.
Let M be a set. Any set map α: U ⊂ Rn → M defined on an open set U of some Rn, n ≥ 0, will be called a plot on M. The name plot is chosen instead of chart to avoid some confusion with the usual notion of chart in a manifold. When possible, a plot α with domain U will be simply denoted by αU.
A diffeology of class C∞ on the set M is any collection P of plots α: Uα ⊂ Rnα → M, nα ≥ 0, verifying the following axioms:
- (1) Any constant map c: Rn → M, n ≥ 0, belongs to P;
- (2) Let α ∈ P be defined on U ⊂ Rn and let h: V ⊂ Rm → U ⊂ Rn beany C∞ map; then α ◦ h ∈ P;
- (3) Let α: U ⊂ Rn → M be a plot. If any t ∈ U has a neighbourhood Ut such that α|Ut belongs to P then α ∈ P.
Usually, a diffeology P on the set M is defined by means of a generating set, that is by giving any set G of plots (which is implicitly supposed to contain all constant maps) and taking the least diffeology containing it. Explicitly, the diffeology ⟨G⟩ generated by G is the set of plots α: U → M such that any point t ∈ U has a neighbourhood Ut where α can be written as γ ◦ h for some C∞ map h and some γ ∈ G.
A finite dimensional manifold M is endowed with the diffeology generated by the charts U ⊂ Rn → M, n = dimM, of any atlas.
Basic constructions. A map F : (M, P) → (N, Q) between diffeological spaces is differentiable if F ◦ α ∈ Q for all α ∈ P. A diffeomorphism is a differentiable map with a differentiable inverse.
Let (M,P) be a diffeological space and F : M → N a map of sets. The final diffeology F⋆P on N is that generated by the plots F ◦ α, α ∈ P. A particular case is the quotient diffeology associated to an equivalence relation on M.
Analogously, let (N, Q) be a diffeological space and F : M → N a map of sets. The initial diffeology F⋆Q on M is that generated by the plots α in M such that F ◦ α ∈ Q. A particular case is the induced diffeology on any subset M ⊂ N.
Finally, let D(M,N) be the space of differentiable maps between two diffeological spaces (M,P) and (N,Q). We define the functional diffeology on it by taking as a generating set all plots α: U → D(M,N) such that the associated map α~ : U × M → N given by α~ (t, x) = α(t)(x) is differentiable.
Definition 2.3. A diffeological group is a diffeological space (G,P) endowed with a group structure such that the division map δ : G × G → G, δ(x, y) = xy−1, is differentiable.
A typical example of diffeological group is the diffeomorphism group of a finite dimensional manifold M, endowed with the diffeology induced by D(M,M). It is proven that the diffeomorphism group of the space of leaves of a Lie foliation is a diffeological group too.
Both constructions verify the usual universal properties.
Let (M, P), (N, Q) be two diffeological spaces. We can endow the cartesian product M × N with the product diffeology P × Q generated by the plots α × β, α ∈ P, β ∈ Q.