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 axioms 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 smooth if and only if its composite with any plot of the first space is a plot of the second space. It is called a diffeomorphism if it is smooth, bijective, and its inverse is also smooth. By construction, given a diffeological space , its plots defined on are precisely all the smooth maps from to .

Diffeological spaces form a category where the morphisms are smooth 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]

D-topology

Any diffeological space is automatically a topological space with the so-called D-topology:[8] the final topology such that all plots are continuous (with respect to the euclidean topology on ).

In other words, a subset is open if and only if is open for any plot on . Actually, the D-topology is completely determined by smooth curves, i.e. a subset is open if and only if is open for any smooth map .[9]

The D-topology is automatically locally path-connected[10] and a differentiable map between diffeological spaces is automatically continuous between their D-topologies.[5]

Additional structures

A Cartan-De Rham calculus can be developed in the framework of diffeologies, 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.[11]

Examples

Trivial examples

Manifolds

Constructions from other diffeological spaces

Wire/spaghetti diffeology

The wire diffeology (or spaghetti diffeology) on is the diffeology whose plots factor locally through . More precisely, a map is a plot if and only if for every there is an open neighbourhood of such that for two plots and . This diffeology does not coincide with the standard diffeology on : for instance, the identity is not a plot in the wire diffeology.[5]

This example can be enlarged to diffeologies whose plots factor locally through . More generally, one can consider the rank--restricted diffeology on a smooth manifold : a map is a plot if and only if the rank of its differential is less or equal than . For one recovers the wire diffeology.[17]

Other 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:

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. For diffeologies underlying smooth manifolds, this boils down to the standard notion of embedding.

References

  1. ^ Souriau, J. M. (1980), García, P. L.; Pérez-Rendón, A.; Souriau, J. M. (eds.), "Groupes differentiels", Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Mathematics, vol. 836, Berlin, Heidelberg: Springer Berlin Heidelberg, 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, Lecture Notes in Physics, vol. 201, Berlin/Heidelberg: Springer-Verlag, 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: ScD thesis, Université de Provence.
  4. ^ Iglesias, Patrick (1985). Fibrés difféologiques et homotopie [Diffeological fiber bundles and homotopy] (PDF) (in French). Marseille: ScD thesis, Université de Provence.
  5. ^ a b c d 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. arXiv:0807.1704. doi:10.1090/S0002-9947-2011-05107-X. ISSN 0002-9947.
  8. ^ Iglesias, Patrick (1985). Fibrés difféologiques et homotopie [Diffeological fiber bundles and homotopy] (PDF) (in French). Marseille: ScD thesis, Université de Provence. Definition 1.2.3
  9. ^ a b Christensen, John Daniel; Sinnamon, Gordon; Wu, Enxin (2014-10-09). "The D -topology for diffeological spaces". Pacific Journal of Mathematics. 272 (1): 87–110. arXiv:1302.2935. doi:10.2140/pjm.2014.272.87. ISSN 0030-8730.
  10. ^ Laubinger, Martin (2006). "Diffeological spaces". Proyecciones. 25 (2): 151–178. doi:10.4067/S0716-09172006000200003. ISSN 0717-6279.
  11. ^ Christensen, Daniel; Wu, Enxin (2016). "Tangent spaces and tangent bundles for diffeological spaces". Cahiers de Topologie et Geométrie Différentielle Catégoriques. 57 (1): 3–50. arXiv:1411.5425.
  12. ^ Iglesias-Zemmour, Patrick; Karshon, Yael; Zadka, Moshe (2010). "Orbifolds as diffeologies" (PDF). Transactions of the American Mathematical Society. 362 (6): 2811–2831. doi:10.1090/S0002-9947-10-05006-3. JSTOR 25677806. S2CID 15210173.
  13. ^ 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.
  14. ^ 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.
  15. ^ 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.
  16. ^ Losik, Mark (1994). "Categorical differential geometry". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 35 (4): 274–290.
  17. ^ Blohmann, Christian (2023-01-06). "Elastic diffeological spaces". arXiv:2301.02583 [math.DG].
  18. ^ Donato, Paul; Iglesias, 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 (in French). 301 (4): 127–130. MR 0799609.