Editing Diffeology (section)

=== 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" />