Editing Diffeology (section)

== 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"/>