# Diffeology

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*^{[1]}^{[2]} and later developed by his students Paul Donato^{[3]} and Patrick Iglesias.^{[4]}^{[5]} 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.^{[6]}

## Intuitive definition[edit]

Recall that a topological manifold is a topological space which is locally homeomorphic to . Differentiable manifolds generalize the notion of smoothness on 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 to the manifold which are used to "pull back" the differential structure from 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 . Indeed, every smooth manifold has a natural diffeology, consisting of its maximal atlas (all the smooth maps from open subsets of to the manifold). This abstract point of view makes no reference to a specific atlas (and therefore to a fixed dimension *) nor to the underlying topological space, and is therefore suitable to treat examples of objects more general than manifolds.
*

## Formal definition[edit]

A **diffeology** on a set * consists of a collection of maps, called ***plots** or parametrizations, from open subsets of (*) to ** 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
*is a plot, and**is a smooth function from an open subset of some real vector space into the domain of**, then the composition**is a plot.*

Note that the domains of different plots can be subsets of for different values of *; in particular, any diffeology contains the elements of its underlying set as the plots with **. 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 , for all *, and open covers.*^{[7]}

### Morphisms[edit]

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.^{[7]}

### Additional structures[edit]

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 ). 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.^{[5]} However, there is not a canonical definition of tangent spaces and tangent bundles for diffeological spaces.^{[8]}

## Examples[edit]

### Manifolds[edit]

- Any differentiable manifold is a diffeological space together with its maximal atlas (i.e., the plots are all smooth maps from open subsets of 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 . For instance, diffeological spaces include orbifolds, which are modeled on quotient spaces , for is a finite linear subgroup,^{[9]}or manifolds with boundary and corners, modeled on orthants, etc.^{[10]} - Any Banach manifold is a diffeological space.
^{[11]} - Any Fréchet manifold is a diffeological space.
^{[12]}^{[13]}

### Constructions from other diffeological spaces[edit]

- If
*is a subset of the diffeological space**, then the***subspace diffeology**on*is the diffeology consisting of the plots of**whose images are subsets of**. The D-topology of**is the subspace topology of the D-topology of**.* - If
*and**are diffeological spaces, then the***product diffeology**on the Cartesian product*is the diffeology generated by all products of plots of**and of**. The D-topology of**is the product topology of the D-topologies of**and**.* - If
*is a diffeological space and**is an equivalence relation on**, then the***quotient diffeology**on the quotient set*/~ is the diffeology generated by all compositions of plots of**with the projection from**to**. The D-topology on**is the quotient topology of the D-topology of**(note that this topology may be trivial without the diffeology being trivial).* - The
**pushforward diffeology**of a diffeological space*by a function**is the diffeology on**generated by the compositions**, for**a plot of**. In other words, the pushforward diffeology is the smallest diffeology on**making**differentiable. The quotient diffeology boils down to the pushforward diffeology by the projection**.* - The
**pullback diffeology**of a diffeological space*by a function**is the diffeology on**whose plots are maps**such that the composition**is a plot of**. In other words, the pullback diffeology is the smallest diffeology on**making**differentiable.* - The
**functional diffeology**between two diffeological spaces is the diffeology on the set of differentiable maps, whose plots are the maps such that is smooth (with respect to the product diffeology of ). When*and**are manifolds, the D-topology of is the smallest locally path-connected topology containing the weak topology.*^{[14]}

### More general examples[edit]

- Any set can be endowed with the
**coarse**(or trivial, or indiscrete)**diffelogy**, 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
**is a smooth manifold. The quotient , for some irrational***, called***irrational torus**, is a diffeological space diffeomorphic to the quotient of the regular 2-torus by a line of slope*. It has a non-trivial diffeology, but its D-topology is the trivial topology.*^{[15]} - 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[edit]

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 * between diffeological spaces such that the diffeology of ** is the pushforward of the diffeology of **. Similarly, an ***induction** is an injective function * between diffeological spaces such that the diffeology of ** *is the pullback of the diffeology of *. Note that subductions and inductions are automatically smooth.
*

When * and ** 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
*is a subduction (respectively, induction) if and only if**is a subduction (respectively,**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.

## References[edit]

**^**Souriau, J. M. (1980), García, P. L.; Pérez-Rendón, A.; Souriau, J. M. (eds.), "Groupes differentiels",*Differential Geometrical Methods in Mathematical Physics*, Berlin, Heidelberg: Springer Berlin Heidelberg,**836**, pp. 91–128, doi:10.1007/bfb0089728, ISBN 978-3-540-10275-5, retrieved 2022-01-16**^**Souriau, Jean-Marie (1984), Denardo, G.; Ghirardi, G.; Weber, T. (eds.), "Groupes différentiels et physique mathématique",*Group Theoretical Methods in Physics*, Berlin/Heidelberg: Springer-Verlag,**201**, pp. 511–513, doi:10.1007/bfb0016198, ISBN 978-3-540-13335-3, retrieved 2022-01-16**^**Donato, Paul (1984).*Revêtement et groupe fondamental des espaces différentiels homogènes*[*Coverings and fundamental groups of homogeneous differential spaces*] (in French). Marseille: PhD thesis, Université de Provence.**^**Iglesias, Patrick (1985).*Fibrés difféologiques et homotopie*[*Diffeological fiber bundles and homotopy*] (PDF) (in French). Marseille: PhD thesis, Université de Provence.- ^
^{a}^{b}Iglesias-Zemmour, Patrick (2013-04-09).*Diffeology*. Mathematical Surveys and Monographs.**185**. American Mathematical Society. doi:10.1090/surv/185. ISBN 978-0-8218-9131-5. **^**Chen, Kuo-Tsai (1977). "Iterated path integrals".*Bulletin of the American Mathematical Society*.**83**(5): 831–879. doi:10.1090/S0002-9904-1977-14320-6. ISSN 0002-9904.- ^
^{a}^{b}Baez, John; Hoffnung, Alexander (2011). "Convenient categories of smooth spaces".*Transactions of the American Mathematical Society*.**363**(11): 5789–5825. doi:10.1090/S0002-9947-2011-05107-X. ISSN 0002-9947. **^**Christensen, Daniel; Wu, Enxin (2016). "Tangent spaces and tangent bundles for diffeological spaces".*Cahiers de Topologie et Geométrie Différentielle Catégoriques*.**57**(1): 3–50.**^**Iglesias-Zemmour, Patrick; Karshon, Yael; Zadka, Moshe (2010). "Orbifolds as diffeologies" (PDF).*Transactions of the American Mathematical Society*.**362**: 2811–2831. JSTOR 25677806.**^**Gürer, Serap; Iglesias-Zemmour, Patrick (2019). "Differential forms on manifolds with boundary and corners".*Indagationes Mathematicae*.**30**(5): 920–929. doi:10.1016/j.indag.2019.07.004.**^**Hain, Richard M. (1979). "A characterization of smooth functions defined on a Banach space".*Proceedings of the American Mathematical Society*.**77**(1): 63–67. doi:10.1090/S0002-9939-1979-0539632-8. ISSN 0002-9939.**^**Losik, Mark (1992). "О многообразиях Фреше как диффеологических пространствах" [Fréchet manifolds as diffeological spaces].*Izv. Vyssh. Uchebn. Zaved. Mat.*(in Russian).**5**: 36–42 – via All-Russian Mathematical Portal.**^**Losik, Mark (1994). "Categorical differential geometry".*Cahiers de Topologie et Géométrie Différentielle Catégoriques*.**35**(4): 274–290.**^**Christensen, John Daniel; Sinnamon, Gordon; Wu, Enxin (2014-10-09). "The D -topology for diffeological spaces".*Pacific Journal of Mathematics*.**272**(1): 87–110. doi:10.2140/pjm.2014.272.87. ISSN 0030-8730.**^**Donato, Paul; Iglésias, Patrick (1985). "Exemples de groupes difféologiques: flots irrationnels sur le tore" [Examples of diffeological groups: irrational flows on the torus].*C. R. Acad. Sci. Paris Sér. I Math.*(in French).**301**(4): 127–130. MR 0799609.

## External links[edit]

- Patrick Iglesias-Zemmour:
*Diffeology*(book), Mathematical Surveys and Monographs, vol. 185, American Mathematical Society, Providence, RI USA [2013]. - Patrick Iglesias-Zemmour: Diffeology (many documents)
- diffeology.net Global hub on diffeology and related topics