In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, by declaring what constitutes the "smooth parametrizations" into the set.
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.
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
A diffeological space consists of a set together with a collection of maps (called a diffeology) satisfying suitable axioms, which are used to characterize smoothness of the space in a way similar to charts of an atlas.
A smooth manifold can be equivalently defined as a diffeological space which is locally diffeomorphic to
But there are many diffeological spaces which do not carry any local model, nor a sufficiently interesting underlying topological space.
Diffeology is therefore suitable to treat examples of objects more general than manifolds.
consists of a collection of maps, called plots or parametrizations, from open subsets of
such that the following axioms hold: Note that the domains of different plots can be subsets of
; in particular, any diffeology contains the elements of its underlying set as the plots with
More abstractly, a diffeological space is a concrete sheaf on the site of open subsets of
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.
Actually, the D-topology is completely determined by smooth curves, i.e. a subset
[9] The D-topology is automatically locally path-connected[10] and a differentiable map between diffeological spaces is automatically continuous between their D-topologies.
[5] 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.
can be assigned the diffeology consisting of all smooth maps from all open subsets of Euclidean spaces into it.
, all constant maps (with domains open subsets of Euclidean spaces), etc.
The D-topology recovers the original manifold topology.
Accordingly, smooth manifolds with smooth maps form a full subcategory of the category of diffeological spaces.
This procedure similarly assigns diffeologies to other spaces that possess a smooth structure that is determined by a local model.
More precisely, each of the examples below form a full subcategory of diffeological spaces.
[5] This example can be enlarged to diffeologies whose plots factor locally through
is a plot if and only if the rank of its differential is less or equal than
[17] Analogously to the notions of submersions and immersions between manifolds, there are two special classes of morphisms between diffeological spaces.
Note that subductions and inductions are automatically smooth.
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.