Editing Diffeology (section)

=== 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"/>