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]

Intuitive definition

Recall that a topological manifold is a topological space which is locally homeomorphic to ${\displaystyle \mathbb {R} ^{n))$. Differentiable manifolds generalize the notion of smoothness on ${\displaystyle \mathbb {R} ^{n))$ 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 ${\displaystyle \mathbb {R} ^{n))$ to the manifold which are used to "pull back" the differential structure from ${\displaystyle \mathbb {R} ^{n))$ 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 ${\displaystyle \mathbb {R} ^{n))$. Indeed, every smooth manifold has a natural diffeology, consisting of its maximal atlas (all the smooth maps from open subsets of ${\displaystyle \mathbb {R} ^{n))$ to the manifold). This abstract point of view makes no reference to a specific atlas (and therefore to a fixed dimension ${\displaystyle n}$) 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 ${\displaystyle X}$ consists of a collection of maps, called plots or parametrizations, from open subsets of ${\displaystyle \mathbb {R} ^{n))$ (${\displaystyle n\geq 0}$) to ${\displaystyle X}$ such that the following axioms hold:

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

Note that the domains of different plots can be subsets of ${\displaystyle \mathbb {R} ^{n))$ for different values of ${\displaystyle n}$; in particular, any diffeology contains the elements of its underlying set as the plots with ${\displaystyle n=0}$. 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 ${\displaystyle \mathbb {R} ^{n))$, for all ${\displaystyle n\geq 0}$, and open covers.^[7]

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 ${\displaystyle X}$, its plots defined on ${\displaystyle U}$ are precisely all the smooth maps from ${\displaystyle U}$ to ${\displaystyle X}$.

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.^[7]

D-topology

Any diffeological space is automatically a topological space with the so-called D-topology:^[8] the final topology such that all plots are continuous (with respect to the euclidean topology on ${\displaystyle \mathbb {R} ^{n))$).

In other words, a subset ${\displaystyle U\subset X}$ is open if and only if ${\displaystyle f^{-1}(U)}$ is open for any plot ${\displaystyle f}$ on ${\displaystyle X}$. Actually, the D-topology is completely determined by smooth curves, i.e. a subset ${\displaystyle U\subset X}$ is open if and only if ${\displaystyle c^{-1}(U)}$ is open for any smooth map ${\displaystyle c:\mathbb {R} \to X}$.^[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]

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.^[5] However, there is not a canonical definition of tangent spaces and tangent bundles for diffeological spaces.^[11]

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 is a diffeological space by considering its maximal atlas (i.e., the plots are all smooth maps from open subsets of ${\displaystyle \mathbb {R} ^{n))$ 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 ${\displaystyle \mathbb {R} ^{n))$. For instance, diffeological spaces include orbifolds, which are modeled on quotient spaces ${\displaystyle \mathbb {R} ^{n}/\Gamma }$, for ${\displaystyle \Gamma }$ is a finite linear subgroup,^[12] or manifolds with boundary and corners, modeled on orthants, etc.^[13]
• Any Banach manifold is a diffeological space.^[14]
• Any Fréchet manifold is a diffeological space.^[15][16]

Constructions from other diffeological spaces

• If a set ${\displaystyle X}$ is given two different diffeologies, their intersection is a diffeology on ${\displaystyle X}$, 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 ${\displaystyle Y}$ is a subset of the diffeological space ${\displaystyle X}$, then the subspace diffeology on ${\displaystyle Y}$ is the diffeology consisting of the plots of ${\displaystyle X}$ whose images are subsets of ${\displaystyle Y}$. The D-topology of ${\displaystyle Y}$ is finer than the subspace topology of the D-topology of ${\displaystyle X}$.
• If ${\displaystyle X}$ and ${\displaystyle Y}$ are diffeological spaces, then the product diffeology on the Cartesian product ${\displaystyle X\times Y}$ is the diffeology generated by all products of plots of ${\displaystyle X}$ and of ${\displaystyle Y}$. The D-topology of ${\displaystyle X\times Y}$ is the product topology of the D-topologies of ${\displaystyle X}$ and ${\displaystyle Y}$.
• If ${\displaystyle X}$ is a diffeological space and ${\displaystyle \sim }$ is an equivalence relation on ${\displaystyle X}$, then the quotient diffeology on the quotient set ${\displaystyle X}$/~ is the diffeology generated by all compositions of plots of ${\displaystyle X}$ with the projection from ${\displaystyle X}$ to ${\displaystyle X/\sim }$. The D-topology on ${\displaystyle X/\sim }$ is the quotient topology of the D-topology of ${\displaystyle X}$ (note that this topology may be trivial without the diffeology being trivial).
• The pushforward diffeology of a diffeological space ${\displaystyle X}$ by a function ${\displaystyle f:X\to Y}$ is the diffeology on ${\displaystyle Y}$ generated by the compositions ${\displaystyle f\circ p}$, for ${\displaystyle p}$ a plot of ${\displaystyle X}$. In other words, the pushforward diffeology is the smallest diffeology on ${\displaystyle Y}$ making ${\displaystyle f}$ differentiable. The quotient diffeology boils down to the pushforward diffeology by the projection ${\displaystyle X\to X/\sim }$.
• The pullback diffeology of a diffeological space ${\displaystyle Y}$ by a function ${\displaystyle f:X\to Y}$ is the diffeology on ${\displaystyle X}$ whose plots are maps ${\displaystyle p}$ such that the composition ${\displaystyle f\circ p}$ is a plot of ${\displaystyle Y}$. In other words, the pullback diffeology is the smallest diffeology on ${\displaystyle X}$ making ${\displaystyle f}$ differentiable.
• The functional diffeology between two diffeological spaces ${\displaystyle X,Y}$ is the diffeology on the set ${\displaystyle {\mathcal {C))^{\infty }(X,Y)}$ of differentiable maps, whose plots are the maps ${\displaystyle \phi :U\to {\mathcal {C))^{\infty }(X,Y)}$ such that ${\displaystyle (u,x)\mapsto \phi (u)(x)}$ is smooth (with respect to the product diffeology of ${\displaystyle U\times X}$). When ${\displaystyle X}$ and ${\displaystyle Y}$ are manifolds, the D-topology of ${\displaystyle {\mathcal {C))^{\infty }(X,Y)}$ is the smallest locally path-connected topology containing the weak topology.^[9]

Wire/spaghetti diffeology

The wire diffeology (or spaghetti diffeology) on ${\displaystyle \mathbb {R} ^{2))$ is the diffeology whose plots factor locally through ${\displaystyle \mathbb {R} }$. More precisely, a map ${\displaystyle p:U\to \mathbb {R} ^{2))$ is a plot if and only if for every ${\displaystyle u\in U}$ there is an open neighbourhood ${\displaystyle V\subseteq U}$ of ${\displaystyle u}$ such that ${\displaystyle p|_{V}=q\circ F}$ for two plots ${\displaystyle F:V\to \mathbb {R} }$ and ${\displaystyle q:\mathbb {R} \to \mathbb {R} ^{2))$. This diffeology does not coincide with the standard diffeology on ${\displaystyle \mathbb {R} ^{2))$: for instance, the identity ${\displaystyle \mathrm {id} :\mathbb {R} ^{2}\to \mathbb {R} ^{2))$ is not a plot in the wire diffeology.^[5]

This example can be enlarged to diffeologies whose plots factor locally through ${\displaystyle \mathbb {R} ^{r))$. More generally, one can consider the rank-${\displaystyle r}$-restricted diffeology on a smooth manifold ${\displaystyle M}$: a map ${\displaystyle U\to M}$ is a plot if and only if the rank of its differential is less or equal than ${\displaystyle r}$. For ${\displaystyle r=1}$ one recovers the wire diffeology.^[17]

Other examples

• Quotients gives an easy way to construct non-manifold diffeologies. For example, the set of real numbers ${\displaystyle \mathbb {R} }$ is a smooth manifold. The quotient ${\displaystyle \mathbb {R} /(\mathbb {Z} +\alpha \mathbb {Z} )}$, for some irrational ${\displaystyle \alpha }$, called irrational torus, is a diffeological space diffeomorphic to the quotient of the regular 2-torus ${\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2))$ by a line of slope ${\displaystyle \alpha }$. It has a non-trivial diffeology, but its D-topology is the trivial topology.^[18]
• 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 ${\displaystyle f:X\to Y}$ between diffeological spaces such that the diffeology of ${\displaystyle Y}$ is the pushforward of the diffeology of ${\displaystyle X}$. Similarly, an induction is an injective function ${\displaystyle f:X\to Y}$ between diffeological spaces such that the diffeology of ${\displaystyle X}$ is the pullback of the diffeology of ${\displaystyle Y}$. Note that subductions and inductions are automatically smooth.

It is instructive to consider the case where ${\displaystyle X}$ and ${\displaystyle Y}$ are smooth manifolds.

• Every surjective submersion ${\displaystyle f:X\to Y}$ is a subduction.
• A subduction need not be a surjective submersion. One example is ${\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }$ given by ${\displaystyle f(x,y):=xy}$.
• An injective immersion need not be an induction. One example is the parametrization of the "figure-eight," ${\displaystyle f:\left(-{\frac {\pi }{2)),{\frac {3\pi }{2))\right)\to \mathbb {R^{2)) }$ given by ${\displaystyle f(t):=(2\cos(t),\sin(2t))}$.
• An induction need not be an injective immersion. One example is the "semi-cubic," ${\displaystyle f:\mathbb {R} \to \mathbb {R} ^{2))$given by ${\displaystyle f(t):=(t^{2},t^{3})}$.^[19][20]

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.^[17]