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Ein diffeologischer Raum ist in der Mathematik, spezieller dem Bereich der Differentialgeometrie, eine starke Verallgemeinerung glatter Mannigfaltigkeiten, vergleichbar mit topologischen Räumen als Verallgemeinerung topologischer Mannigfaltigkeiten. Das Studium diffeologischer Räume heißt analog zur Topologie Diffeologie.

Während in der Topologie die Stetigkeit im Mittelpunkt steht, ist in der Diffeologie die Glattheit ein zentraler Begriff: ein diffeologischer Raum ist im wesentlichen eine Menge $X$ zusammen mit einer Menge von Abbildungen $p:\mathbb {R} ^{n}\to X$ die als glatt betrachtet werden sollen. Diese Struktur ist allgemein genug um viele wichtige Arten von Räumen zu umfassen, wie z.B. glatte Mannigfaltigkeiten auch mit Rand und Ecken, Banach-Mannigfaltigkeiten und Orbifaltigkeiten, aber auch singuläre Räume wie den Raum $C^{\infty }(M,N)$ aller glatten Abbildungen zwischen Mannigfaltigkeiten oder Quotientenräume die sonst keine glatte Struktur tragen. Zugleich bieten diffeologische Räume aber immer noch genug Struktur um viele bekannte Konzepte auf diesen Rahmen zu verallgemeinern, wie zum Beispiel Differentialformen und die De-Rham-Kohomologie.

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:

• 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 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 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.

External links

• Patrick Iglesias-Zemmour: Diffeology (book), Mathematical Surveys and Monographs, vol. 185, American Mathematical Society, Providence, RI USA [2013].
• Patrick Iglesias-Zemmour: Diffeology (many documents)
• diffeology.net Global hub on diffeology and related topics

{{Manifolds}} [[Category:Differential geometry]] [[Category:Functions and mappings]] [[Category:Chinese mathematical discoveries|Chen, Guocai]] [[Category:Smooth manifolds]]