# Diffeology

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 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 (also called smooth 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 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 $\mathbb {R} ^{n}$. 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.

## Formal definition

A diffeology on a set $X$ consists of a collection of maps, called plots or parametrizations, from open subsets of $\mathbb {R} ^{n}$ (for all $n\geq 0$) to $X$ such that the following axioms hold:
* 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.
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.
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.

## Examples
### 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 $M$ can be assigned the diffeology consisting of all smooth maps from all open subsets of Euclidean spaces into it. This diffeology will contain not only the charts of $M$, but also all smooth curves into $M$, all constant maps (with domains open subsets of Euclidean spaces), etc. The D-topology recovers the original manifold topology. With this diffeology, a map between two smooth manifolds is smooth in the usual sense if and only if it is smooth in the diffeological sense. 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.
* Orbifolds, which are modeled on quotient spaces $\mathbb {R} ^{n}/\Gamma$, for $\Gamma$ is a finite linear subgroup, and smooth maps between them.
* Manifolds with boundary or corners, which are modeled on orthants, and smooth maps between them.
* Banach manifolds, which are modeled on Banach spaces, and smooth maps between them.
* Fréchet manifolds, which are modeled on Fréchet spaces, and smooth maps between them.

### 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 equal to the subspace topology of the D-topology of $X$ if $Y$ is open, but may be finer in general.
* 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 coarsest delta-generated topology containing the product topology of the D-topologies of $X$ and $Y$; it is equal to the product topology when $X$ or $Y$ is locally compact, but may be finer in general.
* 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.