Editing Diffeology

{{Distinguish|Diffiety}}

In [[mathematics]], a '''diffeology''' on a set generalizes the concept of a smooth atlas of a [[differentiable manifold]], by declaring only what constitutes the "smooth parametrizations" into the set. A diffeological space is a set equipped with a diffeology. Many of the standard tools of [[differential geometry]] extend to diffeological spaces, which beyond manifolds include arbitrary quotients of manifolds, arbitrary subsets of manifolds, and spaces of mappings between manifolds.

== Introduction ==
=== Calculus on "smooth spaces" ===
The [[differential calculus]] on <math>\mathbb{R}^n</math>, or, more generally, on finite dimensional [[vector space]]s, is one of the most impactful successes of modern mathematics. Fundamental to its basic definitions and theorems is the linear structure of the underlying space.<ref name="Spiv65"/><ref name="Mun91"/>

The field of [[differential geometry]] establishes and studies the extension of the classical differential calculus to non-linear spaces. This extension is made possible by the definition of a [[differentiable manifold|smooth manifold]], which is also the starting point for diffeological spaces.

A smooth <math>n</math>-dimensional manifold is a set <math>M</math> equipped with a maximal [[smooth atlas]], which consists of injective functions, called [[Chart (mathematics)|charts]], of the form <math>\phi:U \to M</math>, where <math>U</math> is an open subset of <math>\mathbb{R}^n</math>, satisfying some mutual-compatibility relations. The charts of a manifold perform two distinct functions, which are often syncretized:<ref name="KobNom96"/><ref name="Tu11"/><ref name="Lee13"/>
* They dictate the local structure of the manifold. The chart <math>\phi:U \to M</math> identifies its image in <math>M</math> with its domain <math>U</math>. This is convenient because the latter is simply an open subset of a [[Euclidean space]].
* They define the class of smooth maps between manifolds. These are the maps to which the differential calculus extends. In particular, the charts determine smooth functions (smooth maps <math>M \to \mathbb{R}</math>), smooth [[curve]]s (smooth maps <math>\mathbb{R} \to M</math>), smooth [[Homotopy|homotopies]] (smooth maps <math>\mathbb{R}^2 \to M</math>), etc.

A diffeology generalizes the structure of a smooth manifold by abandoning the first requirement for an atlas, namely that the charts give a local model of the space, while retaining the ability to discuss smooth maps into the space.<ref name="Igl13"/><ref name="Igl21" /><ref name="Igl22" />

=== Informal definition ===
A '''diffeological space''' is a set <math>X</math> equipped with a '''diffeology''': a collection of maps<math display="block">\{p:U \to X\mid U \text{ is an open subset of }\mathbb{R}^n, \text{ and } n \geq 0\},</math>whose members are called '''plots''', that satisfies some axioms. The plots are not required to be injective, and can (indeed, must) have as domains the open subsets of arbitrary Euclidean spaces.

A smooth manifold can be viewed as a diffeological space which is locally diffeomorphic to <math>\mathbb{R}^n</math>. In general, while not giving local models for the space, the axioms of a diffeology still ensure that the plots induce a coherent notion of smooth functions, smooth curves, smooth homotopies, etc. Diffeology is therefore suitable to treat objects more general than manifolds.<ref name="Igl13"/><ref name="Igl21"/><ref name="Igl22"/>

=== Motivating example ===
Let <math>M</math> and <math>N</math> be smooth manifolds. A smooth homotopy of maps <math>M \to N</math> is a smooth map <math>H:\mathbb{R} \times M \to N</math>. For each <math>t \in \mathbb{R}</math>, the map <math>H_t := H(t, \cdot):M \to N</math> is smooth, and the intuition behind a smooth homotopy is that it is a smooth curve into the space of smooth functions <math>\mathcal{C}^\infty(M,N)</math> connecting, say, <math>H_0</math> and <math>H_1</math>. But <math>\mathcal{C}^\infty(M,N)</math> is not a finite-dimensional smooth manifold, so formally we cannot yet speak of smooth curves into it.

On the other hand, the collection of maps
<math display="block">\{p:U \to \mathcal{C}^\infty(M,N) \mid \text{ the map }U \times M \to N, \ (r,x) \mapsto p(r)(x) \text{ is smooth}\}</math>
is a diffeology on <math>\mathcal{C}^\infty(M,N)</math>. With this structure, the smooth curves (a notion which is now rigorously defined) correspond precisely to the smooth homotopies.<ref name="Igl13"/><ref name="Igl21"/><ref name="Igl22"/>

=== History ===
The concept of diffeology was first introduced by [[Jean-Marie Souriau]] in the 1980s under the name ''espace différentiel.''<ref name="Sour80"/><ref name="Sour84"/> Souriau's motivating application for diffeology was to uniformly handle the infinite-dimensional groups arising from his work in [[geometric quantization]]. Thus the notion of diffeological group preceded the more general concept of a diffeological space. Souriau's diffeological program was taken up by his students, particularly [[Paul G. Donato|Paul Donato]]<ref name="Don84"/> and [[Patrick Iglesias-Zemmour]],<ref name="Igl85"/> who completed early pioneering work in the field.

A structure similar to diffeology was introduced by [[Kuo-Tsaï Chen]] (陳國才, ''Chen Guocai'') in the 1970s, in order to formalize certain computations with path integrals. Chen's definition used [[convex set]]s instead of open sets for the domains of the plots.<ref name="Chen77"/> The similarity between diffeological and "Chen" structures can be made precise by viewing both as concrete sheaves over the appropriate concrete site.<ref name="BaezHof11"/>

== Formal definition ==
A '''diffeology''' on a set ''<math>X</math>'' consists of a collection of maps, called '''plots''' or parametrizations, from [[Open set|open subsets]] of <math>\mathbb{R}^n</math> (for all ''<math>n \geq 0</math>'') to ''<math>X</math>'' such that the following axioms hold:

* '''Covering axiom''': every constant map is a plot.
* '''Locality axiom''': for a given map ''<math>p: U \to X</math>'', if every point in ''<math>U</math>'' has a [[Neighborhood (topology)|neighborhood]] ''<math>V \subset U</math>'' such that ''<math>p|_V</math>'' is a plot, then ''<math>p</math>'' itself is a plot.
* '''Smooth compatibility axiom''': if ''<math>p</math>'' is a plot, and ''<math>F</math>'' is a [[smooth function]] from an open subset of some <math>\mathbb{R}^m</math> into the domain of ''<math>p</math>'', then the composite ''<math>p \circ F</math>'' is a plot.

Note that the domains of different plots can be subsets of <math>\mathbb{R}^n</math> for different values of ''<math>n</math>''; in particular, any diffeology contains the elements of its underlying set as the plots with ''<math>n = 0</math>''. A set together with a diffeology is called a '''diffeological space'''.

More abstractly, a diffeological space is a concrete [[Sheaf (mathematics)|sheaf]] on the [[Site (mathematics)|site]] of open subsets of <math>\mathbb{R}^n</math>, for all ''<math>n \geq 0</math>'', and [[open cover]]s.<ref name="BaezHof11"/>

=== 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 function|inverse]] is also smooth. Equipping the open subsets of Euclidean spaces with their standard diffeology (as defined in the next section), the plots into a diffeological space ''<math>X</math>'' are precisely the smooth maps from ''<math>U</math>'' to ''<math>X</math>''.

Diffeological spaces constitute the objects of a [[Category theory|category]], denoted by <math>\mathsf{Dflg}</math>, whose [[morphism]]s are smooth maps. The category <math>\mathsf{Dflg}</math> is closed under many categorical operations: for instance, it is [[Cartesian closed category|Cartesian closed]], [[Complete category|complete]] and [[Cocomplete category|cocomplete]], and more generally it is a [[quasitopos]].<ref name="BaezHof11"/>

=== D-topology ===
Any diffeological space is a [[topological space]] when equipped with the '''D-topology''':<ref name="Igl85" /> the [[final topology]] such that all plots are [[Continuous function (topology)|continuous]] (with respect to the [[Euclidean topology]] on <math>\mathbb{R}^n</math>).

In other words, a subset <math>U \subset X</math> is open if and only if <math>p^{-1}(U)</math> is open for any plot <math>p</math> on <math>X</math>. Actually, the D-topology is completely determined by smooth [[curve]]s, i.e. a subset <math>U \subset X</math> is open if and only if <math>c^{-1}(U)</math> is open for any smooth map <math>c: \mathbb{R} \to X</math>.<ref name="ChrSinWu14"/> The D-topology is automatically [[Locally path connected|locally path-connected]]<ref name="Laub06"/>

A smooth map between diffeological spaces is automatically [[Continuous function|continuous]] between their D-topologies.<ref name="Igl13" /> Therefore we have the functor <math>D:\mathsf{Dflg} \to \mathsf{Top}</math>, from the category of diffeological spaces to the category of topological spaces, which assigns to a diffeological space its D-topology. This functor realizes <math>\mathsf{Dflg}</math> as a [[concrete category]] over <math>\mathsf{Top}</math>.

=== 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 bundle]]s, [[homotopy]], etc.<ref name="Igl13"/> However, there is not a canonical definition of [[tangent space]]s and [[tangent bundle]]s for diffeological spaces.<ref name="ChrWu14"/>

== Examples ==

=== First examples ===
Any set carries at least two diffeologies:
* the '''coarse''' (or trivial, or indiscrete) diffeology, consisting of every map into the set. This is the largest possible diffeology. The corresponding D-topology is the [[trivial topology]].
* the '''discrete''' (or fine) diffeology, consisting of the locally constant maps into the set. This is the smallest possible diffeology. The corresponding D-topology is the [[discrete topology]].

Any topological space can be endowed with the '''continuous''' diffeology, whose plots are the [[Continuous function|continuous]] maps.

The Euclidean space <math>\mathbb{R}^n</math>admits several diffeologies beyond those listed above.

* The '''standard''' diffeology on <math>\mathbb{R}^n</math> consists of those maps <math>p:U \to \mathbb{R}^n</math> which are smooth in the usual sense of multivariable calculus.
* The '''wire''' (or spaghetti) diffeology on <math>\mathbb{R}^n</math> is the diffeology whose plots factor locally through <math>\mathbb{R}</math>. More precisely, a map <math>p: U \to \mathbb{R}^n</math> is a plot if and only if for every <math>u \in U</math> there is an open neighbourhood <math>V \subseteq U</math> of <math>u</math> such that <math>p|_V = q \circ F</math> for two smooth functions <math>F: V \to \mathbb{R}</math> and <math>q: \mathbb{R} \to \mathbb{R}^n</math>. This diffeology does not coincide with the standard diffeology on <math>\mathbb{R}^n</math> when <math>n\geq 2</math>: for instance, the identity <math display="inline">\mathbb{R}^n \to X= \mathbb{R}^n</math> is not a plot for the wire diffeology.<ref name="Igl13"/>
* The previous example can be enlarged to diffeologies whose plots factor locally through <math>\mathbb{R}^r</math>, yielding the '''rank-<math>r</math>-restricted''' diffeology on a smooth manifold <math>M</math>: a map <math>U \to M</math> is a plot if and only if it is smooth and the rank of its [[Pushforward (differential)|differential]] is less than or equal than <math>r</math>. For <math>r=1</math> one recovers the wire diffeology.<ref name="Bloh24"/>

=== Relation to other smooth spaces ===
Diffeological spaces generalize manifolds, but they are far from the only mathematical objects to do so. For instance manifolds with corners, orbifolds, and infinite-dimensional Fréchet manifolds are all well-established alternatives. This subsection makes precise the extent to which these spaces are diffeological.

We view <math>\mathsf{Dflg}</math> as a concrete category over the category of topological spaces <math>\mathsf{Top}</math> via the D-topology functor <math>D:\mathsf{Dflg} \to \mathsf{Top}</math>. If <math>U:\mathsf{C} \to \mathsf{Top}</math> is another concrete category over <math>\mathsf{Top}</math>, we say that a functor <math>E:\mathsf{C} \to \mathsf{Dflg}</math> is an embedding (of concrete categories) if it is injective on objects and faithful, and <math>D \circ E = U</math>. To specify an embedding, we need only describe it on objects; it is necessarily the identity map on arrows.

We will say that a diffeological space <math>X</math> is '''locally modeled''' by a collection of diffeological spaces <math>\mathcal{E}</math> if around every point <math>x \in X</math>, there is a D-open neighbourhood <math>U</math>, a D-open subset <math>V</math> of some <math>E \in \mathcal{E}</math>, and a diffeological diffeomorphism <math>U \to V</math>.<ref name="Igl13"/><ref name="Nest21"/>

==== Manifolds ====
The category of finite-dimensional smooth manifolds (allowing those with connected components of different dimensions) fully embeds into <math>\mathsf{Dflg}</math>. The embedding <math>y</math> assigns to a smooth manifold <math>M</math> the canonical diffeology<math display="block">\{p:U \to M \mid p \text{ is smooth in the usual sense}\}.</math>In particular, a diffeologically smooth map between manifolds is smooth in the usual sense, and the D-topology of <math>y(M)</math> is the original topology of <math>M</math>. The [[Image (category theory)#Essential Image|essential image]] of this embedding consists of those diffeological spaces that are locally modeled by the collection <math>\{y(\mathbb{R}^n)\}</math>, and whose D-topology is [[Hausdorff space|Hausdorff]] and [[Second-countable space|second-countable]].<ref name="Igl13"/>

==== Manifolds with boundary or corners ====
The category of finite-dimensional smooth [[Manifold#Manifold with boundary|manifolds with boundary]] (allowing those with connected components of different dimensions) similarly fully embeds into <math>\mathsf{Dflg}</math>. The embedding is defined identically to the smooth case, except "smooth in the usual sense" refers to the standard definition of smooth maps between manifolds with boundary. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection <math>\{y(O) \mid O \text{ is a half-space}\}</math>, and whose D-topology is Hausdorff and second-countable. The same can be done in more generality for [[Manifold with corners|manifolds with corners]], using the collection <math>\{y(O) \mid O \text{ is an orthant}\}</math>.<ref name="GurIgl19"/>

==== Fréchet and Banach manifolds ====
The category of [[Fréchet manifold]]s similarly fully embeds into <math>\mathsf{Dflg}</math>. Once again, the embedding is defined identically to the smooth case, except "smooth in the usual sense" refers to the standard definition of smooth maps between Fréchet spaces. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection <math>\{y(E) \mid E \text{ is a Fréchet space}\}</math>, and whose D-topology is Hausdorff.

The embedding restricts to one of the category of [[Banach manifold]]s. Historically, the case of Banach manifolds was proved first, by Hain,<ref name="Hain79"/> and the case of Fréchet manifolds was treated later, by Losik.<ref name="Los92"/><ref name="Los94"/> The category of manifolds modeled on [[convenient vector space]]s also similarly embeds into <math>\mathsf{Dflg}</math>.<ref name="FrolKrieg88"/><ref name="Miy25"/>

==== Orbifolds ====
A (classical) [[orbifold]] <math>X</math> is a space that is locally modeled by quotients of the form <math>\mathbb{R}^n/\Gamma</math>, where <math>\Gamma</math> is a [[Finite group|finite subgroup]] of linear transformations. On the other hand, each model <math>\mathbb{R}^n/\Gamma</math> is naturally a diffeological space (with the quotient diffeology discussed below), and therefore the orbifold charts generate a diffeology on <math>X</math>. This diffeology is uniquely determined by the orbifold structure of <math>X</math>.

Conversely, a diffeological space that is locally modeled by the collection <math>\{\mathbb{R}^n/\Gamma\}</math> (and with Hausdorff D-topology) carries a classical orbifold structure that induces the original diffeology, wherein the local diffeomorphisms are the orbifold charts. Such a space is called a diffeological orbifold.<ref name="IglKarZad10"/>

Whereas diffeological orbifolds automatically have a notion of smooth map between them (namely diffeologically smooth maps in <math>\mathsf{Dflg}</math>), the notion of a smooth map between classical orbifolds is not standardized.

If orbifolds are viewed as [[differentiable stack]]s presented by étale proper [[Lie groupoid]]s, then there is a functor from the underlying 1-category of orbifolds, and equivalent maps-of-stacks between them, to <math>\mathsf{Dflg}</math>. Its essential image consists of diffeological orbifolds, but the functor is neither faithful nor full.<ref name="Miy24"/>

== Constructions ==
=== Intersections ===
If a set ''<math>X</math>'' is given two different diffeologies, their [[intersection]] is a diffeology on ''<math>X</math>'', called the '''intersection diffeology''', which is finer than both starting diffeologies. The D-topology of the intersection diffeology is finer than the intersection of the D-topologies of the original diffeologies.

=== Products ===
If ''<math>X</math>'' and ''<math>Y</math>'' are diffeological spaces, then the '''product''' diffeology on the [[Cartesian product]] ''<math>X \times Y</math>'' is the diffeology generated by all products of plots of ''<math>X</math>'' and of ''<math>Y</math>''. Precisely, a map <math>p:U \to X \times Y</math> necessarily has the form <math>p(u) = (x(u),y(u))</math> for maps <math>x:U \to X</math> and <math>y:U \to Y</math>. The map <math>p</math> is a plot in the product diffeology if and only if <math>x</math> and <math>y</math> are plots of <math>X</math> and <math>Y</math>, respectively. This generalizes to products of arbitrary collections of spaces.

The D-topology of ''<math>X \times Y</math>'' is the coarsest delta-generated topology containing the [[product topology]] of the D-topologies of ''<math>X</math>'' and ''<math>Y</math>''; it is equal to the product topology when ''<math>X</math>'' or ''<math>Y</math>'' is [[locally compact]], but may be finer in general.<ref name="ChrSinWu14"/>

=== Pullbacks ===
Given a map ''<math>f: X \to Y</math>'' from a set <math>X</math> to a diffeological space <math>Y</math>, the '''pullback''' diffeology on ''<math>X</math>'' consists of those maps ''<math>p:U \to X</math>'' such that the composition ''<math>f \circ p</math>'' is a plot of ''<math>Y</math>''. In other words, the pullback diffeology is the smallest diffeology on ''<math>X</math>'' making ''<math>f</math>'' smooth.

If ''<math>X</math>'' is a [[subset]] of the diffeological space ''<math>Y</math>'', then the '''subspace''' diffeology on ''<math>X</math>'' is the pullback diffeology induced by the inclusion <math>X \hookrightarrow Y</math>. In this case, the D-topology of ''<math>X</math>'' is equal to the [[subspace topology]] of the D-topology of ''<math>Y</math>'' if ''<math>Y</math>'' is open, but may be finer in general.

=== Pushforwards ===
Given a map ''<math>f: X \to Y</math>'' from diffeological space ''<math>X</math>'' to a set <math>Y</math>, the '''pushforward''' diffeology on ''<math>Y</math>'' is the diffeology generated by the compositions ''<math>f \circ p</math>'', for plots ''<math>p:U \to X</math>'' of ''<math>X</math>''. In other words, the pushforward diffeology is the smallest diffeology on ''<math>Y</math>'' making ''<math>f</math>'' smooth.

If ''<math>X</math>'' is a diffeological space and ''<math>\sim</math>'' is an [[equivalence relation]] on ''<math>X</math>'', then the '''quotient''' diffeology on the [[quotient set]] ''<math>X/{\sim}</math>'' is the pushforward diffeology induced by the quotient map <math>X \to X/{\sim}</math>. The D-topology on ''<math>X/{\sim}</math>'' is the [[quotient topology]] of the D-topology of ''<math>X</math>''. Note that this topology may be trivial without the diffeology being trivial.

Quotients often give rise to non-manifold diffeologies. For example, the set of [[real number]]s '''<math>\mathbb{R}</math>''' is a smooth manifold. The quotient <math>\mathbb{R}/(\mathbb{Z} + \alpha \mathbb{Z})</math>, for some [[Irrational number|irrational]] ''<math>\alpha</math>'', called the '''irrational torus''', is a diffeological space diffeomorphic to the quotient of the regular [[Torus|2-torus]] <math>\mathbb{R}^2/\mathbb{Z}^2</math> by a line of [[slope]] ''<math>\alpha</math>''. It has a non-trivial diffeology, although its D-topology is the [[trivial topology]].<ref name="DonIgl85"/>

=== Functional diffeologies ===
The '''functional''' diffeology on the set <math>\mathcal{C}^{\infty}(X,Y)</math> of smooth maps between two diffeological spaces <math>X</math> and <math>Y</math> is the diffeology whose plots are the maps <math>\phi: U \to \mathcal{C}^{\infty}(X,Y)</math> such that<math display="block">U \times X \to Y, \quad (u,x) \mapsto \phi(u)(x)</math>is smooth with respect to the product diffeology of <math>U \times X</math>. When ''<math>X</math>'' and ''<math>Y</math>'' are manifolds, the D-topology of <math>\mathcal{C}^{\infty}(X,Y)</math> is the smallest [[Locally path connected|locally path-connected]] topology containing the [[Whitney Topologies|Whitney <math>C^\infty</math> topology]].<ref name="ChrSinWu14"/>

Taking the subspace diffeology of a functional diffeology, one can define diffeologies on the space of [[Section (fiber bundle)|sections]] of a [[Fiber bundle|fibre bundle]], or the space of bisections of a [[Lie groupoid]], etc.

If <math>M</math> is a compact smooth manifold, and <math>F \to M</math> is a smooth fiber bundle over <math>M</math>, then the space of smooth sections <math>\Gamma(F)</math> of the bundle is frequently equipped with the structure of a Fréchet manifold.<ref name="Ham82"/> Upon embedding this Fréchet manifold into the category of diffeological spaces, the resulting diffeology coincides with the subspace diffeology that <math>\Gamma(F)</math> inherits from the functional diffeology on <math>\mathcal{C}^\infty(M,F)</math>.<ref name="Wal12"/>

== Distinguished maps between diffeological spaces ==
Analogous to the notions of [[Submersion (mathematics)|submersions]] and [[Immersion (mathematics)|immersions]] between manifolds, there are two special classes of morphisms between diffeological spaces. A '''subduction''' is a surjective function ''<math>f: X \to Y</math>'' between diffeological spaces such that the diffeology of ''<math>Y</math>'' is the pushforward of the diffeology of ''<math>X</math>''. Similarly, an '''induction''' is an injective function ''<math>f: X \to Y</math>'' between diffeological spaces such that the diffeology of ''<math>X</math> ''is the pullback of the diffeology of ''<math>Y</math>''. Subductions and inductions are automatically smooth.

It is instructive to consider the case where ''<math>X</math>'' and ''<math>Y</math>'' are smooth manifolds.

* Every surjective [[Submersion (mathematics)|submersion]] <math>f:X \to Y</math> is a subduction.
* A subduction need not be a surjective submersion. One example is <math display="block">f:\mathbb{R}^2 \to \mathbb{R}, \quad f(x,y) := xy.</math>
* An injective [[Immersion (mathematics)|immersion]] need not be an induction. One example is the parametrization of the "figure-eight,"
<math display="block">f:\left(-\frac{\pi}{2}, \frac{3\pi}{2}\right) \to \mathbb{R^2}, \quad f(t) := (2\cos(t), \sin(2t)).</math>
* An induction need not be an injective immersion. One example is the "semi-cubic,"<ref name="KarMiyWat24"/><ref name="Jor82"/>
<math display="block">f:\mathbb{R} \to \mathbb{R}^2, \quad f(t) := (t^2, t^3).</math>

In the category of diffeological spaces, subductions are precisely the strong [[epimorphism]]s, and inductions are precisely the strong [[monomorphism]]s.<ref name="Bloh24"/> A map that is both a subduction and induction is a diffeomorphism.

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}}

==External links==
* Patrick Iglesias-Zemmour: [http://math.huji.ac.il/~piz/ Diffeology (many documents)]
* [http://diffeology.net/ 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]]