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.
If X is a set, a diffeology on X is a set of maps, called plots, from open subsets of Rn (n ≥ 0) to X such that the following hold:
- Every constant map is a plot.
- For a given map, if every point in the domain has a neighborhood such that restricting the map to this neighborhood is a plot, then the map itself is a plot.
- If p is a plot, and f is a smooth function from an open subset of some real vector space into the domain of p, then the composition p ∘ f is a plot.
Note that the domains of different plots can be subsets of Rn for different values of n.
A set together with a diffeology is called a diffeological space.
A map between diffeological spaces is called differentiable if and only if composing it with every plot of the first space is a plot of the second space. It is a diffeomorphism if it is differentiable, bijective, and its inverse is also differentiable.
The diffeological spaces, together with differentiable maps as morphisms, form a category. The isomorphisms in this category are just the diffeomorphisms defined above. The category of diffeological spaces is closed under many categorical operations.
If Y is a subset of the diffeological space X, then Y is itself a diffeological space in a natural way: the plots of Y are those plots of X whose images are subsets of Y.
If X is a diffeological space and ~ is some equivalence relation on X, then the quotient set X/~ has the diffeology generated by all compositions of plots of X with the projection from X to X/~. This is called the quotient diffeology. Note that the quotient D-topology is the D-topology of the quotient diffeology, and that this topology may be trivial without the diffeology being trivial.
A Cartan De Rham calculus can be developed in the framework of diffeology, as well as fiber bundles, homotopy etc.
Differentiable manifolds also generalize smoothness. They are normally defined as topological manifolds with an atlas, whose transition maps are smooth, which is used to pull back the differential structure.
Every smooth manifold defined in this way has a natural diffeology, for which the plots correspond to the smooth maps from open subsets of Rn to the manifold. With this diffeology, a map between two smooth manifolds is smooth if and only if it is differentiable in the diffeological sense. Hence the smooth manifolds with smooth maps form a full subcategory of the diffeological spaces.
This allows one to give an alternative definition of smooth manifold which makes no reference to transition maps or to a specific atlas: a smooth manifold is a diffeological space which is locally diffeomorphic to Rn.
The relationship between smooth manifolds and diffeological spaces is analogous to the relationship between topological manifolds and topological spaces.
This method of modeling diffeological spaces can be extended to other locals models, for instance: orbifolds, modeled on quotient spaces Rn/Γ where Γ is a finite linear subgroup, or manifolds with boundary and corners, modeled on orthants etc.
- Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.
- Any smooth manifold is a diffeological space.
- Any quotient of a diffeological space is a diffeological space. This is an easy way to construct non-manifold diffeologies. For example, the set of real numbers R is a smooth manifold. The quotient R/(Z + αZ), for some irrational α, is the irrational torus, a diffeological space diffeomorphic to the quotient of the regular 2-torus R2/Z2 by a line of slope α. It has a non-trivial diffeology, but its D-topology is the trivial topology.