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.
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 axioms hold:
- Covering axiom: every constant map is a plot.
- Locality axiom: for a given map , if every point in has a neighborhood such that is a plot, then itself is a plot.
- Smooth compatibility axiom: if is a plot, and is a smooth function from an open subset of some into the domain of , then the composite is a plot.
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.
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 , its plots defined on are precisely all the smooth maps from to .
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.
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 is open if and only if is open for any plot on . Actually, the D-topology is completely determined by smooth curves, i.e. a subset is open if and only if 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.
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.
- Any set can be endowed with the coarse (or trivial, or indiscrete) diffeology, i.e. the largest possible diffeology (any map is a plot). The corresponding D-topology is the trivial topology.
- Any set can be endowed with the discrete (or fine) diffeology, i.e. the smallest possible diffeology (the only plots are the locally constant maps). The corresponding D-topology is the discrete topology.
- Any topological space can be endowed with the continuous diffeology, whose plots are all continuous maps.
- Any differentiable manifold is a diffeological space by considering its maximal atlas (i.e., the plots are all smooth maps from open subsets of to the manifold); its D-topology recovers the original manifold topology. With this diffeology, a map between two smooth manifolds is smooth if and only if it is differentiable in the diffeological sense. Accordingly, smooth manifolds with smooth maps form a full subcategory of the category of diffeological spaces.
- Similarly, complex manifolds, analytic manifolds, etc. have natural diffeologies consisting of the maps preserving the extra structure.
- This method of modeling diffeological spaces can be extended to locals models which are not necessarily the euclidean space . For instance, diffeological spaces include orbifolds, which are modeled on quotient spaces , for is a finite linear subgroup, or manifolds with boundary and corners, modeled on orthants, etc.
- Any Banach manifold is a diffeological space.
- Any Fréchet manifold is a diffeological space.
Constructions from other diffeological spaces
- If a set is given two different diffeologies, their intersection is a diffeology on , called the intersection diffeology, which is finer than both starting diffeologies. The D-topology of the intersection diffeology is the intersection of the D-topologies of the initial diffeologies.
- If is a subset of the diffeological space , then the subspace diffeology on is the diffeology consisting of the plots of whose images are subsets of . The D-topology of is the subspace topology of the D-topology of .
- If and are diffeological spaces, then the product diffeology on the Cartesian product is the diffeology generated by all products of plots of and of . The D-topology of is the product topology of the D-topologies of and .
- If is a diffeological space and is an equivalence relation on , then the quotient diffeology on the quotient set /~ is the diffeology generated by all compositions of plots of with the projection from to . The D-topology on is the quotient topology of the D-topology of (note that this topology may be trivial without the diffeology being trivial).
- The pushforward diffeology of a diffeological space by a function is the diffeology on generated by the compositions , for a plot of . In other words, the pushforward diffeology is the smallest diffeology on making differentiable. The quotient diffeology boils down to the pushforward diffeology by the projection .
- The pullback diffeology of a diffeological space by a function is the diffeology on whose plots are maps such that the composition is a plot of . In other words, the pullback diffeology is the smallest diffeology on making differentiable.
- The functional diffeology between two diffeological spaces is the diffeology on the set of differentiable maps, whose plots are the maps such that is smooth (with respect to the product diffeology of ). When and are manifolds, the D-topology of is the smallest locally path-connected topology containing the weak topology.
The wire diffeology (or spaghetti diffeology) on is the diffeology whose plots factor locally through . More precisely, a map is a plot if and only if for every there is an open neighbourhood of 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--restricted diffeology on a smooth manifold : a map is a plot if and only if the rank of its differential is less or equal than . For one recovers the wire diffeology.
- Quotients gives an easy way to construct non-manifold diffeologies. For example, the set of real numbers is a smooth manifold. The quotient , for some irrational , called irrational torus, is a diffeological space diffeomorphic to the quotient of the regular 2-torus by a line of slope . It has a non-trivial diffeology, but its D-topology is the trivial topology.
- Combining the subspace diffeology and the functional diffeology, one can define diffeologies on the space of sections of a fibre bundle, or the space of bisections of a Lie groupoid, etc.
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 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:
- A composition is a subduction (respectively, induction) if and only if is a subduction (respectively, is an induction).
- An injective subduction (respectively, a surjective induction) is a diffeomorphism.
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. For diffeologies underlying smooth manifolds, this boils down to the standard notion of embedding.