Diffeology

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 $$\mathbb{R}^n$$. Differentiable manifolds generalize the notion of smoothness on $$\mathbb{R}^n$$ 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 $$\mathbb{R}^n$$ to the manifold which are used to "pull back" the differential structure from $$\mathbb{R}^n$$ 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 $$\mathbb{R}^n$$. Indeed, every smooth manifold has a natural diffeology, consisting of its maximal atlas (all the smooth maps from open subsets of $$\mathbb{R}^n$$ 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 $$\mathbb{R}^n$$ ($$n \geq 0$$) to $$X$$ such that the following axioms hold:

Note that the domains of different plots can be subsets of $$\mathbb{R}^n$$ 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.
 * Covering axiom: every constant map is a plot.
 * Locality axiom: for a given map $$f: U \to X$$, if every point in $$U$$ has a neighborhood $$V \subset U$$ such that $$f_{\mid V}$$ is a plot, then $$f$$ itself is a plot.
 * Smooth compatibility axiom: if $$p$$ is a plot, and $$f$$ is a smooth function from an open subset of some $$\mathbb{R}^m$$ into the domain of $$p$$, then the composite $$p \circ f$$ is a plot.

More abstractly, a diffeological space is a concrete sheaf on the site of open subsets of $$\mathbb{R}^n$$, 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 $$\mathbb{R}^n$$).

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 $$c: \mathbb{R} \to X$$.

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.

Trivial examples

 * 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.

Manifolds

 * Any differentiable manifold is a diffeological space by considering its maximal atlas (i.e., the plots are all smooth maps from open subsets of $$\mathbb{R}^n$$ 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 $$\mathbb{R}^n$$. For instance, diffeological spaces include orbifolds, which are modeled on quotient spaces $$\mathbb{R}^n/\Gamma$$, for $$\Gamma$$ 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 $$X$$ is given two different diffeologies, their intersection is a diffeology on $$X$$, 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 $$Y$$ is a subset of the diffeological space $$X$$, then the subspace diffeology on $$Y$$ is the diffeology consisting of the plots of $$X$$ whose images are subsets of $$Y$$. The D-topology of $$Y$$ is finer than the subspace topology of the D-topology of $$X$$.
 * If $$X$$ and $$Y$$ are diffeological spaces, then the product diffeology on the Cartesian product $$X \times Y$$ is the diffeology generated by all products of plots of $$X$$ and of $$Y$$. The D-topology of $$X \times Y$$ is the product topology of the D-topologies of $$X$$ and $$Y$$.
 * If $$X$$ is a diffeological space and $$\sim$$ is an equivalence relation on $$X$$, then the quotient diffeology on the quotient set $$X$$/~ is the diffeology generated by all compositions of plots of $$X$$ with the projection from $$X$$ to $$X/\sim$$. The D-topology on $$X/\sim$$ is the quotient topology of the D-topology of $$X$$ (note that this topology may be trivial without the diffeology being trivial).
 * The pushforward diffeology of a diffeological space $$X$$ by a function $$f: X \to Y$$ is the diffeology on $$Y$$ generated by the compositions $$f \circ p$$, for $$p$$ a plot of $$X$$. In other words, the pushforward diffeology is the smallest diffeology on $$Y$$ making $$f$$ differentiable. The quotient diffeology boils down to the pushforward diffeology by the projection $$X \to X/\sim$$.
 * The pullback diffeology of a diffeological space $$Y$$ by a function $$f: X \to Y$$ is the diffeology on $$X$$ whose plots are maps $$p$$ such that the composition $$f \circ p$$ is a plot of $$Y$$. In other words, the pullback diffeology is the smallest diffeology on $$X$$ making $$f$$ differentiable.
 * The functional diffeology between two diffeological spaces $$X,Y$$ is the diffeology on the set $$\mathcal{C}^{\infty}(X,Y)$$ of differentiable maps, whose plots are the maps $$\phi: U \to \mathcal{C}^{\infty}(X,Y)$$ such that $$(u,x) \mapsto \phi(u)(x)$$ is smooth (with respect to the product diffeology of $$U \times X$$). When $$X$$ and $$Y$$ are manifolds, the D-topology of $$\mathcal{C}^{\infty}(X,Y)$$ is the smallest locally path-connected topology containing the weak topology.

Wire/spaghetti diffeology
The wire diffeology (or spaghetti diffeology) on $$\mathbb{R}^2$$ is the diffeology whose plots factor locally through $$\mathbb{R}$$. More precisely, a map $$p: U \to \mathbb{R}^2$$ is a plot if and only if for every $$u \in U$$ there is an open neighbourhood $$V \subseteq U$$ of $$u$$ such that $$p|_V = q \circ F$$ for two plots $$F: V \to \mathbb{R}$$ and $$q: \mathbb{R} \to \mathbb{R}^2$$. This diffeology does not coincide with the standard diffeology on $$\mathbb{R}^2$$: for instance, the identity $$\mathrm{id}: \mathbb{R}^2 \to \mathbb{R}^2$$ is not a plot in the wire diffeology.

This example can be enlarged to diffeologies whose plots factor locally through $$\mathbb{R}^r$$. 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

 * Quotients gives an easy way to construct non-manifold diffeologies. For example, the set of real numbers $$\mathbb{R}$$ is a smooth manifold. The quotient $$\mathbb{R}/(\mathbb{Z} + \alpha \mathbb{Z})$$, for some irrational $$\alpha$$, called irrational torus, is a diffeological space diffeomorphic to the quotient of the regular 2-torus $$\mathbb{R}^2/\mathbb{Z}^2$$ by a line of slope $$\alpha$$. 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 $$f: X \to Y$$ 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.


 * Every surjective submersion $$f:X \to Y$$ is a subduction.
 * A subduction need not be a surjective submersion. One example is $$f:\mathbb{R}^2 \to \mathbb{R}$$ given by $$f(x,y) := xy$$.
 * An injective immersion need not be an induction. One example is the parametrization of the "figure-eight," $$f:\left(-\frac{\pi}{2}, \frac{3\pi}{2}\right) \to \mathbb{R^2}$$ given by $$f(t) := (2\cos(t), \sin(2t))$$.
 * An induction need not be an injective immersion. One example is the "semi-cubic," $$f:\mathbb{R} \to \mathbb{R}^2$$given by $$f(t) := (t^2, t^3)$$.

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.