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[1][2] and later developed by his students Paul Donato[3] and Patrick Iglesias.[4][5] 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.[6]
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 ) nor to the underlying topological space, and is therefore suitable to treat examples of objects more general than manifolds.
A diffeology on a set consists of a collection of maps, called plots or parametrizations, from open subsets of () to such that the following properties hold:
Note that the domains of different plots can be subsets of for different values of ; in particular, any diffeology contains the elements of its underlying set as the plots with . 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 , and open covers.[7]
A map between diffeological spaces is called differentiable (or smooth) if and only if its composition with any plot of the first space is a plot of the second space. It is called a diffeomorphism if it is differentiable, bijective, and its inverse is also differentiable.
Diffeological spaces form a category, whose morphisms are differentiable 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.[7]
Any diffeological space is automatically a topological space with the so-called D-topology: the finest topology such that all plots are continuous (with respect to the euclidean topology on ). A differentiable map between diffeological spaces is automatically continuous between their D-topologies.
A Cartan-De Rham calculus can be developed in the framework of diffeology, as well as a suitable adaptation of the notions of fiber bundles, homotopy, etc.[5] However, there is not a canonical definition of tangent spaces and tangent bundles for diffeological spaces.[8]
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 between diffeological spaces such that the diffeology of is the pushforward of the diffeology of . Similarly, an induction is an injective function between diffeological spaces such that the diffeology of is the pullback of the diffeology of . Note that subductions and inductions are automatically smooth.
When and are smooth manifolds, a subduction (respectively, induction) between them is precisely a surjective submersion (respectively, injective immersion). Moreover, these notions enjoy similar properties to submersion and immersions, such as:
Last, an embedding is an induction which is also a homeomorphism with its image, with respect to the subset topology induced from the D-topology of the codomain. This boils down to the standard notion of embedding between manifolds.