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## Summary

In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are.

The concept was first introduced by Jean-Marie Souriau in the 1980s under the name Espace différentiel and later developed by his students Paul Donato and Patrick Iglesias. A related idea was introduced by Kuo-Tsaï Chen (陳國才, Chen Guocai) in the 1970s, using convex sets instead of open sets for the domains of the plots.

## Intuitive definition

Recall that a topological manifold is a topological space which is locally homeomorphic to $\mathbb {R} ^{n}$ . Differentiable manifolds generalize the notion of smoothness on $\mathbb {R} ^{n}$  in the following sense: a differentiable manifold is a topological manifold with a differentiable atlas, i.e. a collection of maps from open subsets of $\mathbb {R} ^{n}$  to the manifold which are used to "pull back" the differential structure from $\mathbb {R} ^{n}$  to the manifold.

A diffeological space consists of a set together with a collection of maps (called a diffeology) satisfying suitable axioms, which generalise the notion of an atlas on a manifold. In this way, the relationship between smooth manifolds and diffeological spaces is analogous to the relationship between topological manifolds and topological spaces:

More precisely, a smooth manifold can be equivalently defined as a diffeological space which is locally diffeomorphic to $\mathbb {R} ^{n}$ . Indeed, every smooth manifold has a natural diffeology, consisting of its maximal atlas (all the smooth maps from open subsets of $\mathbb {R} ^{n}$  to the manifold). This abstract point of view makes no reference to a specific atlas (and therefore to a fixed dimension $n$ ) nor to the underlying topological space, and is therefore suitable to treat examples of objects more general than manifolds.

## Formal definition

A diffeology on a set $X$  consists of a collection of maps, called plots or parametrizations, from open subsets of $\mathbb {R} ^{n}$  ($n\geq 0$ ) to $X$  such that the following properties hold:

• Every constant map is a plot.
• For a given map, if every point in the domain has a neighborhood such that restricting the map to this neighborhood is a plot, then the map itself is a plot.
• If $p$  is a plot, and $f$  is a smooth function from an open subset of some real vector space into the domain of $p$ , then the composition $p\circ f$  is a plot.

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

More abstractly, a diffeological space is a concrete sheaf on the site of open subsets of $\mathbb {R} ^{n}$ , for all $n\geq 0$ , and open covers.

### Morphisms

A map between diffeological spaces is called differentiable (or smooth) if and only if its composition with any plot of the first space is a plot of the second space. It is called a diffeomorphism if it is differentiable, bijective, and its inverse is also differentiable.

Diffeological spaces form a category, whose morphisms are differentiable maps. The category of diffeological spaces is closed under many categorical operations: for instance, it is Cartesian closed, complete and cocomplete, and more generally it is a quasitopos.

Any diffeological space is automatically a topological space with the so-called D-topology: the finest topology such that all plots are continuous (with respect to the euclidean topology on $\mathbb {R} ^{n}$ ). A differentiable map between diffeological spaces is automatically continuous between their D-topologies.

A Cartan-De Rham calculus can be developed in the framework of diffeology, as well as a suitable adaptation of the notions of fiber bundles, homotopy, etc. However, there is not a canonical definition of tangent spaces and tangent bundles for diffeological spaces.

## Examples

### Manifolds

• Any differentiable manifold is a diffeological space together with its maximal atlas (i.e., the plots are all smooth maps from open subsets of $\mathbb {R} ^{n}$  to the manifold); its D-topology recovers the original manifold topology. With this diffeology, a map between two smooth manifolds is smooth if and only if it is differentiable in the diffeological sense. Accordingly, smooth manifolds with smooth maps form a full subcategory of the category of diffeological spaces.
• Similarly, complex manifolds, analytic manifolds, etc. have natural diffeologies consisting of the maps preserving the extra structure.
• This method of modeling diffeological spaces can be extended to locals models which are not necessarily the euclidean space $\mathbb {R} ^{n}$ . For instance, diffeological spaces include orbifolds, which are modeled on quotient spaces $\mathbb {R} ^{n}/\Gamma$ , for $\Gamma$  is a finite linear subgroup, or manifolds with boundary and corners, modeled on orthants, etc.
• Any Banach manifold is a diffeological space.
• Any Fréchet manifold is a diffeological space.

### Constructions from other diffeological spaces

• If $Y$  is a subset of the diffeological space $X$ , then the subspace diffeology on $Y$  is the diffeology consisting of the plots of $X$  whose images are subsets of $Y$ . The D-topology of $Y$  is the subspace topology of the D-topology of $X$ .
• If $X$  and $Y$  are diffeological spaces, then the product diffeology on the Cartesian product $X\times Y$  is the diffeology generated by all products of plots of $X$  and of $Y$ . The D-topology of $X\times Y$  is the product topology of the D-topologies of $X$  and $Y$ .
• If $X$  is a diffeological space and $\sim$  is an equivalence relation on $X$ , then the quotient diffeology on the quotient set $X$ /~ is the diffeology generated by all compositions of plots of $X$  with the projection from $X$  to $X/\sim$ . The D-topology on $X/\sim$  is the quotient topology of the D-topology of $X$  (note that this topology may be trivial without the diffeology being trivial).
• The pushforward diffeology of a diffeological space $X$  by a function $f:X\to Y$  is the diffeology on $Y$  generated by the compositions $f\circ p$ , for $p$  a plot of $X$ . In other words, the pushforward diffeology is the smallest diffeology on $Y$  making $f$  differentiable. The quotient diffeology boils down to the pushforward diffeology by the projection $X\to X/\sim$ .
• The pullback diffeology of a diffeological space $Y$  by a function $f:X\to Y$  is the diffeology on $X$  whose plots are maps $p$  such that the composition $f\circ p$  is a plot of $X$ . In other words, the pullback diffeology is the smallest diffeology on $X$  making $f$  differentiable.
• The functional diffeology between two diffeological spaces $X,Y$  is the diffeology on the set ${\mathcal {C}}^{\infty }(X,Y)$  of differentiable maps, whose plots are the maps $\phi :U\to {\mathcal {C}}^{\infty }(X,Y)$  such that $(u,x)\mapsto \phi (u)(x)$  is smooth (with respect to the product diffeology of $U\times X$ ). When $X$  and $Y$  are manifolds, the D-topology of ${\mathcal {C}}^{\infty }(X,Y)$  is the smallest locally path-connected topology containing the weak topology.

### More general examples

• Any set can be endowed with the coarse (or trivial, or indiscrete) diffeology, i.e. the largest possible diffeology (any map is a plot). The corresponding D-topology is the trivial topology.
• Any set can be endowed with the discrete (or fine) diffeology, i.e. the smallest possible diffeology (the only plots are the locally constant maps). The corresponding D-topology is the discrete topology.
• Any topological space can be endowed with the continuous diffeology, whose plots are all continuous maps. The corresponding D-topology is of course the original topology of the space.
• Quotients gives an easy way to construct non-manifold diffeologies. For example, the set of real numbers $\mathbb {R}$  is a smooth manifold. The quotient $\mathbb {R} /(\mathbb {Z} +\alpha \mathbb {Z} )$ , for some irrational $\alpha$ , called irrational torus, is a diffeological space diffeomorphic to the quotient of the regular 2-torus $\mathbb {R} ^{2}/\mathbb {Z} ^{2}$  by a line of slope $\alpha$ . It has a non-trivial diffeology, but its D-topology is the trivial topology.
• Combining the subspace diffeology and the functional diffeology, one can define diffeologies on the space of sections of a fibre bundle, or the space of bisections of a Lie groupoid, etc.

## Subductions and inductions

Analogously to the notions of submersions and immersions between manifolds, there are two special classes of morphisms between diffeological spaces. A subduction is a surjective function $f:X\to Y$  between diffeological spaces such that the diffeology of $Y$  is the pushforward of the diffeology of $X$ . Similarly, an induction is an injective function $f:X\to Y$  between diffeological spaces such that the diffeology of $X$  is the pullback of the diffeology of $Y$ . Note that subductions and inductions are automatically smooth.

When $X$  and $Y$  are smooth manifolds, a subduction (respectively, induction) between them is precisely a surjective submersion (respectively, injective immersion). Moreover, these notions enjoy similar properties to submersion and immersions, such as:

• A composition $f\circ g$  is a subduction (respectively, induction) if and only if $f$  is a subduction (respectively, $g$  is an induction).
• An injective subduction (respectively, a surjective induction) is a diffeomorphism.

Last, an embedding is an induction which is also a homeomorphism with its image, with respect to the subset topology induced from the D-topology of the codomain. This boils down to the standard notion of embedding between manifolds.