In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are. The concept was first introduced by Jean-Marie Souriau in the 1980s under the name Espace différentiel and later developed by his students Paul Donato and Patrick Iglesias. 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.

Intuitive definition

Recall that a topological manifold is a topological space which is locally homeomorphic to. Differentiable manifolds generalize the notion of smoothness on in the following sense: a differentiable manifold is a topological manifold with a differentiable atlas, i.e. a collection of maps from open subsets of to the manifold which are used to "pull back" the differential structure from to the manifold. A diffeological space consists of a set together with a collection of maps (called a diffeology) satisfying suitable axioms, which generalise the notion of an atlas on a manifold. In this way, the relationship between smooth manifolds and diffeological spaces is analogous to the relationship between topological manifolds and topological spaces. More precisely, a smooth manifold can be equivalently defined as a diffeological space which is locally diffeomorphic to. Indeed, every smooth manifold has a natural diffeology, consisting of its maximal atlas (all the smooth maps from open subsets of to the manifold). This abstract point of view makes no reference to a specific atlas (and therefore to a fixed dimension n) nor to the underlying topological space, and is therefore suitable to treat examples of objects more general than manifolds.

Formal definition

A diffeology on a set X consists of a collection of maps, called plots or parametrizations, from open subsets of (n \geq 0) to X such that the following axioms hold: Note that the domains of different plots can be subsets of for different values of n; in particular, any diffeology contains the elements of its underlying set as the plots with n = 0. A set together with a diffeology is called a diffeological space. More abstractly, a diffeological space is a concrete sheaf on the site of open subsets of, for all n \geq 0, and open covers.

Morphisms

A map between diffeological spaces is called smooth if and only if its composite with any plot of the first space is a plot of the second space. It is called a diffeomorphism if it is smooth, bijective, and its inverse is also smooth. By construction, given a diffeological space X, its plots defined on U are precisely all the smooth maps from U to X. Diffeological spaces form a category where the morphisms are smooth maps. The category of diffeological spaces is closed under many categorical operations: for instance, it is Cartesian closed, complete and cocomplete, and more generally it is a quasitopos.

D-topology

Any diffeological space is automatically a topological space with the so-called D-topology: the final topology such that all plots are continuous (with respect to the euclidean topology on ). In other words, a subset U \subset X is open if and only if f^{-1}(U) is open for any plot f on X. Actually, the D-topology is completely determined by smooth curves, i.e. a subset U \subset X is open if and only if c^{-1}(U) is open for any smooth map. The D-topology is automatically locally path-connected and a differentiable map between diffeological spaces is automatically continuous between their D-topologies.

Additional structures

A Cartan-De Rham calculus can be developed in the framework of diffeologies, as well as a suitable adaptation of the notions of fiber bundles, homotopy, etc. However, there is not a canonical definition of tangent spaces and tangent bundles for diffeological spaces.

Examples

Trivial examples

Manifolds

Constructions from other diffeological spaces

Wire/spaghetti diffeology

The wire diffeology (or spaghetti diffeology) on is the diffeology whose plots factor locally through \mathbb{R}. More precisely, a map is a plot if and only if for every u \in U there is an open neighbourhood of u such that for two plots and. This diffeology does not coincide with the standard diffeology on : for instance, the identity is not a plot in the wire diffeology. This example can be enlarged to diffeologies whose plots factor locally through. More generally, one can consider the rank-r-restricted diffeology on a smooth manifold M: a map U \to M is a plot if and only if the rank of its differential is less or equal than r. For r=1 one recovers the wire diffeology.

Other examples

Subductions and inductions

Analogously to the notions of submersions and immersions between manifolds, there are two special classes of morphisms between diffeological spaces. A subduction is a surjective function f: X \to Y between diffeological spaces such that the diffeology of Y is the pushforward of the diffeology of X. Similarly, an induction is an injective function f: X \to Y between diffeological spaces such that the diffeology of X is the pullback of the diffeology of Y. Note that subductions and inductions are automatically smooth. It is instructive to consider the case where X and Y are smooth manifolds. In the category of diffeological spaces, subductions are precisely the strong epimorphisms, and inductions are precisely the strong monomorphisms. A map that is both a subduction and induction is a diffeomorphism.