Editing Diffeology (section)

== 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"/>