Diffeological space

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

* Every constant map is a plot.
* For a given map, if every point in the domain has a neighborhood such that restricting the map to this neighborhood is a plot, then the map itself is a plot.
* If ''$p$'' is a plot, and ''$f$'' is a smooth function from an open subset of some real vector space into the domain of ''$p$'', then the composition ''$p \circ f$'' is a plot.
Note that the domains of different plots can be subsets of $\mathbb^n$ 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 $\mathbb^n$, for all ''$n \geq 0$'', and open covers.

Morphisms

A map between diffeological spaces is called differentiable (or smooth) if and only if its composition with any plot of the first space is a plot of the second space. It is called a diffeomorphism if it is differentiable, bijective, and its inverse is also differentiable.

Diffeological spaces form a category, whose morphisms are differentiable 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.

Additional structures

Any diffeological space is automatically a topological space with the so-called D-topology: the finest topology such that all plots are continuous (with respect to the euclidean topology on $\mathbb^n$). 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 diffeology, 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

Manifolds

* Any differentiable manifold is a diffeological space together with its maximal atlas (i.e., the plots are all smooth maps from open subsets of $\mathbb^n$ 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 $\mathbb^n$. For instance, diffeological spaces include orbifolds, which are modeled on quotient spaces $\mathbb^n/\Gamma$, 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 ''$Y$'' 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 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 $\mathcal^(X,Y)$ of differentiable maps, whose plots are the maps $\phi: U \to \mathcal^(X,Y)$ 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 $\mathcal^(X,Y)$ is the smallest locally path-connected topology containing the weak topology.

More general 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.
* Quotients gives an easy way to construct non-manifold diffeologies. For example, the set of real numbers $\mathbb$ is a smooth manifold. The quotient $\mathbb/(\mathbb + \alpha \mathbb)$, for some irrational ''$\alpha$'', called irrational torus, is a diffeological space diffeomorphic to the quotient of the regular 2-torus $\mathbb^2/\mathbb^2$ 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.

When ''$X$'' and ''$Y$'' 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 ''$f \circ g$'' is a subduction (respectively, induction) if and only if ''$f$'' is a subduction (respectively, ''$g$'' is an induction).
* An injective subduction (respectively, a surjective induction) is a diffeomorphism.

Last, 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. This boils down to the standard notion of embedding between manifolds.

References

External links

* Patrick Iglesias-Zemmour
''Diffeology''
(book), Mathematical Surveys and Monographs, vol. 185, American Mathematical Society, Providence, RI USA 013
* Patrick Iglesias-Zemmour
Diffeology (many documents)

diffeology.net
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