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

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

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:

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

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

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

Manifolds

Constructions from other diffeological spaces

More general examples

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 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:

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

  1. ^ 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, vol. 836, pp. 91–128, doi:10.1007/bfb0089728, ISBN 978-3-540-10275-5, retrieved 2022-01-16
  2. ^ 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, vol. 201, pp. 511–513, doi:10.1007/bfb0016198, ISBN 978-3-540-13335-3, retrieved 2022-01-16
  3. ^ 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.
  4. ^ Iglesias, Patrick (1985). Fibrés difféologiques et homotopie [Diffeological fiber bundles and homotopy] (PDF) (in French). Marseille: PhD thesis, Université de Provence.
  5. ^ a b Iglesias-Zemmour, Patrick (2013-04-09). Diffeology. Mathematical Surveys and Monographs. Vol. 185. American Mathematical Society. doi:10.1090/surv/185. ISBN 978-0-8218-9131-5.
  6. ^ 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.
  7. ^ 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.
  8. ^ Christensen, Daniel; Wu, Enxin (2016). "Tangent spaces and tangent bundles for diffeological spaces". Cahiers de Topologie et Geométrie Différentielle Catégoriques. 57 (1): 3–50.
  9. ^ Iglesias-Zemmour, Patrick; Karshon, Yael; Zadka, Moshe (2010). "Orbifolds as diffeologies" (PDF). Transactions of the American Mathematical Society. 362: 2811–2831. JSTOR 25677806.
  10. ^ 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.
  11. ^ 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.
  12. ^ 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.
  13. ^ Losik, Mark (1994). "Categorical differential geometry". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 35 (4): 274–290.
  14. ^ 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.
  15. ^ 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.