Diffeology

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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 [math]\displaystyle{ \mathbb{R}^n }[/math]. Differentiable manifolds generalize the notion of smoothness on [math]\displaystyle{ \mathbb{R}^n }[/math] 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 [math]\displaystyle{ \mathbb{R}^n }[/math] to the manifold which are used to "pull back" the differential structure from [math]\displaystyle{ \mathbb{R}^n }[/math] 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 [math]\displaystyle{ \mathbb{R}^n }[/math]. Indeed, every smooth manifold has a natural diffeology, consisting of its maximal atlas (all the smooth maps from open subsets of [math]\displaystyle{ \mathbb{R}^n }[/math] to the manifold). This abstract point of view makes no reference to a specific atlas (and therefore to a fixed dimension [math]\displaystyle{ n }[/math]) 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 [math]\displaystyle{ X }[/math] consists of a collection of maps, called plots or parametrizations, from open subsets of [math]\displaystyle{ \mathbb{R}^n }[/math] ([math]\displaystyle{ n \geq 0 }[/math]) to [math]\displaystyle{ X }[/math] such that the following axioms hold:

  • Covering axiom: every constant map is a plot.
  • Locality axiom: for a given map [math]\displaystyle{ f: U \to X }[/math], if every point in [math]\displaystyle{ U }[/math] has a neighborhood [math]\displaystyle{ V \subset U }[/math] such that [math]\displaystyle{ f_{\mid V} }[/math] is a plot, then [math]\displaystyle{ f }[/math] itself is a plot.
  • Smooth compatibility axiom: if [math]\displaystyle{ p }[/math] is a plot, and [math]\displaystyle{ f }[/math] is a smooth function from an open subset of some [math]\displaystyle{ \mathbb{R}^m }[/math] into the domain of [math]\displaystyle{ p }[/math], then the composite [math]\displaystyle{ p \circ f }[/math] is a plot.

Note that the domains of different plots can be subsets of [math]\displaystyle{ \mathbb{R}^n }[/math] for different values of [math]\displaystyle{ n }[/math]; in particular, any diffeology contains the elements of its underlying set as the plots with [math]\displaystyle{ n = 0 }[/math]. 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 [math]\displaystyle{ \mathbb{R}^n }[/math], for all [math]\displaystyle{ n \geq 0 }[/math], 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 [math]\displaystyle{ X }[/math], its plots defined on [math]\displaystyle{ U }[/math] are precisely all the smooth maps from [math]\displaystyle{ U }[/math] to [math]\displaystyle{ X }[/math].

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 [math]\displaystyle{ \mathbb{R}^n }[/math]).

In other words, a subset [math]\displaystyle{ U \subset X }[/math] is open if and only if [math]\displaystyle{ f^{-1}(U) }[/math] is open for any plot [math]\displaystyle{ f }[/math] on [math]\displaystyle{ X }[/math]. Actually, the D-topology is completely determined by smooth curves, i.e. a subset [math]\displaystyle{ U \subset X }[/math] is open if and only if [math]\displaystyle{ c^{-1}(U) }[/math] is open for any smooth map [math]\displaystyle{ c: \mathbb{R} \to X }[/math].[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

  • 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.

Manifolds

  • Any differentiable manifold is a diffeological space by considering its maximal atlas (i.e., the plots are all smooth maps from open subsets of [math]\displaystyle{ \mathbb{R}^n }[/math] 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 [math]\displaystyle{ \mathbb{R}^n }[/math]. For instance, diffeological spaces include orbifolds, which are modeled on quotient spaces [math]\displaystyle{ \mathbb{R}^n/\Gamma }[/math], for [math]\displaystyle{ \Gamma }[/math] is a finite linear subgroup,[12] or manifolds with boundary and corners, modeled on orthants, etc.[13]
  • Any Banach manifold is a diffeological space.[14]
  • Any Fréchet manifold is a diffeological space.[15][16]

Constructions from other diffeological spaces

  • If a set [math]\displaystyle{ X }[/math] is given two different diffeologies, their intersection is a diffeology on [math]\displaystyle{ X }[/math], called the intersection diffeology, which is finer than both starting diffeologies. The D-topology of the intersection diffeology is the intersection of the D-topologies of the initial diffeologies.
  • If [math]\displaystyle{ Y }[/math] is a subset of the diffeological space [math]\displaystyle{ X }[/math], then the subspace diffeology on [math]\displaystyle{ Y }[/math] is the diffeology consisting of the plots of [math]\displaystyle{ X }[/math] whose images are subsets of [math]\displaystyle{ Y }[/math]. The D-topology of [math]\displaystyle{ Y }[/math] is the subspace topology of the D-topology of [math]\displaystyle{ X }[/math].
  • If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are diffeological spaces, then the product diffeology on the Cartesian product [math]\displaystyle{ X \times Y }[/math] is the diffeology generated by all products of plots of [math]\displaystyle{ X }[/math] and of [math]\displaystyle{ Y }[/math]. The D-topology of [math]\displaystyle{ X \times Y }[/math] is the product topology of the D-topologies of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math].
  • If [math]\displaystyle{ X }[/math] is a diffeological space and [math]\displaystyle{ \sim }[/math] is an equivalence relation on [math]\displaystyle{ X }[/math], then the quotient diffeology on the quotient set [math]\displaystyle{ X }[/math]/~ is the diffeology generated by all compositions of plots of [math]\displaystyle{ X }[/math] with the projection from [math]\displaystyle{ X }[/math] to [math]\displaystyle{ X/\sim }[/math]. The D-topology on [math]\displaystyle{ X/\sim }[/math] is the quotient topology of the D-topology of [math]\displaystyle{ X }[/math] (note that this topology may be trivial without the diffeology being trivial).
  • The pushforward diffeology of a diffeological space [math]\displaystyle{ X }[/math] by a function [math]\displaystyle{ f: X \to Y }[/math] is the diffeology on [math]\displaystyle{ Y }[/math] generated by the compositions [math]\displaystyle{ f \circ p }[/math], for [math]\displaystyle{ p }[/math] a plot of [math]\displaystyle{ X }[/math]. In other words, the pushforward diffeology is the smallest diffeology on [math]\displaystyle{ Y }[/math] making [math]\displaystyle{ f }[/math] differentiable. The quotient diffeology boils down to the pushforward diffeology by the projection [math]\displaystyle{ X \to X/\sim }[/math].
  • The pullback diffeology of a diffeological space [math]\displaystyle{ Y }[/math] by a function [math]\displaystyle{ f: X \to Y }[/math] is the diffeology on [math]\displaystyle{ X }[/math] whose plots are maps [math]\displaystyle{ p }[/math] such that the composition [math]\displaystyle{ f \circ p }[/math] is a plot of [math]\displaystyle{ Y }[/math]. In other words, the pullback diffeology is the smallest diffeology on [math]\displaystyle{ X }[/math] making [math]\displaystyle{ f }[/math] differentiable.
  • The functional diffeology between two diffeological spaces [math]\displaystyle{ X,Y }[/math] is the diffeology on the set [math]\displaystyle{ \mathcal{C}^{\infty}(X,Y) }[/math] of differentiable maps, whose plots are the maps [math]\displaystyle{ \phi: U \to \mathcal{C}^{\infty}(X,Y) }[/math] such that [math]\displaystyle{ (u,x) \mapsto \phi(u)(x) }[/math] is smooth (with respect to the product diffeology of [math]\displaystyle{ U \times X }[/math]). When [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are manifolds, the D-topology of [math]\displaystyle{ \mathcal{C}^{\infty}(X,Y) }[/math] is the smallest locally path-connected topology containing the weak topology.[9]

Wire/spaghetti diffeology

The wire diffeology (or spaghetti diffeology) on [math]\displaystyle{ \mathbb{R}^2 }[/math] is the diffeology whose plots factor locally through [math]\displaystyle{ \mathbb{R} }[/math]. More precisely, a map [math]\displaystyle{ p: U \to \mathbb{R}^2 }[/math] is a plot if and only if for every [math]\displaystyle{ u \in U }[/math] there is an open neighbourhood [math]\displaystyle{ V \subseteq U }[/math] of [math]\displaystyle{ u }[/math] such that [math]\displaystyle{ p|_V = q \circ F }[/math] for two plots [math]\displaystyle{ F: V \to \mathbb{R} }[/math] and [math]\displaystyle{ q: \mathbb{R} \to \mathbb{R}^2 }[/math]. This diffeology does not coincide with the standard diffeology on [math]\displaystyle{ \mathbb{R}^2 }[/math]: for instance, the identity [math]\displaystyle{ \mathrm{id}: \mathbb{R}^2 \to \mathbb{R}^2 }[/math] is not a plot in the wire diffeology.[5]

This example can be enlarged to diffeologies whose plots factor locally through [math]\displaystyle{ \mathbb{R}^r }[/math]. More generally, one can consider the rank-[math]\displaystyle{ r }[/math]-restricted diffeology on a smooth manifold [math]\displaystyle{ M }[/math]: a map [math]\displaystyle{ U \to M }[/math] is a plot if and only if the rank of its differential is less or equal than [math]\displaystyle{ r }[/math]. For [math]\displaystyle{ r=1 }[/math] one recovers the wire diffeology.[17]

Other examples

  • Quotients gives an easy way to construct non-manifold diffeologies. For example, the set of real numbers [math]\displaystyle{ \mathbb{R} }[/math] is a smooth manifold. The quotient [math]\displaystyle{ \mathbb{R}/(\mathbb{Z} + \alpha \mathbb{Z}) }[/math], for some irrational [math]\displaystyle{ \alpha }[/math], called irrational torus, is a diffeological space diffeomorphic to the quotient of the regular 2-torus [math]\displaystyle{ \mathbb{R}^2/\mathbb{Z}^2 }[/math] by a line of slope [math]\displaystyle{ \alpha }[/math]. It has a non-trivial diffeology, but its D-topology is the trivial topology.[18]
  • 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 [math]\displaystyle{ f: X \to Y }[/math] between diffeological spaces such that the diffeology of [math]\displaystyle{ Y }[/math] is the pushforward of the diffeology of [math]\displaystyle{ X }[/math]. Similarly, an induction is an injective function [math]\displaystyle{ f: X \to Y }[/math] between diffeological spaces such that the diffeology of [math]\displaystyle{ X }[/math] is the pullback of the diffeology of [math]\displaystyle{ Y }[/math]. Note that subductions and inductions are automatically smooth.

When [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] 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 [math]\displaystyle{ f \circ g }[/math] is a subduction (respectively, induction) if and only if [math]\displaystyle{ f }[/math] is a subduction (respectively, [math]\displaystyle{ g }[/math] is an induction).
  • An injective subduction (respectively, a surjective induction) is a diffeomorphism.

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 (Berlin, Heidelberg: Springer Berlin Heidelberg) 836: pp. 91–128, doi:10.1007/bfb0089728, ISBN 978-3-540-10275-5, http://link.springer.com/10.1007/BFb0089728, retrieved 2022-01-16 
  2. ↑ Souriau, Jean-Marie (1984), Denardo, G.; Ghirardi, G.; Weber, T., eds., "Groupes différentiels et physique mathématique" (in en), Group Theoretical Methods in Physics, Lecture Notes in Physics (Berlin/Heidelberg: Springer-Verlag) 201: pp. 511–513, doi:10.1007/bfb0016198, ISBN 978-3-540-13335-3, http://link.springer.com/10.1007/BFb0016198, retrieved 2022-01-16 
  3. ↑ Donato, Paul (1984) (in fr). Revêtement et groupe fondamental des espaces différentiels homogènes. Marseille: ScD thesis, Université de Provence. 
  4. ↑ Iglesias, Patrick (1985) (in fr). Fibrés difféologiques et homotopie. Marseille: ScD thesis, Université de Provence. https://math.huji.ac.il/~piz/documents/TheseEtatPI.pdf. 
  5. ↑ 5.0 5.1 5.2 5.3 Iglesias-Zemmour, Patrick (2013-04-09) (in en). Diffeology. Mathematical Surveys and Monographs. 185. American Mathematical Society. doi:10.1090/surv/185. ISBN 978-0-8218-9131-5. https://www.ams.org/surv/185. 
  6. ↑ Chen, Kuo-Tsai (1977). "Iterated path integrals" (in en). Bulletin of the American Mathematical Society 83 (5): 831–879. doi:10.1090/S0002-9904-1977-14320-6. ISSN 0002-9904. https://www.ams.org/bull/1977-83-05/S0002-9904-1977-14320-6/. 
  7. ↑ 7.0 7.1 Baez, John; Hoffnung, Alexander (2011). "Convenient categories of smooth spaces" (in en). Transactions of the American Mathematical Society 363 (11): 5789–5825. doi:10.1090/S0002-9947-2011-05107-X. ISSN 0002-9947. https://www.ams.org/tran/2011-363-11/S0002-9947-2011-05107-X/. 
  8. ↑ Iglesias, Patrick (1985) (in fr). Fibrés difféologiques et homotopie. Marseille: ScD thesis, Université de Provence. https://math.huji.ac.il/~piz/documents/TheseEtatPI.pdf. "Definition 1.2.3" 
  9. ↑ 9.0 9.1 Christensen, John Daniel; Sinnamon, Gordon; Wu, Enxin (2014-10-09). "The D -topology for diffeological spaces" (in en). Pacific Journal of Mathematics 272 (1): 87–110. doi:10.2140/pjm.2014.272.87. ISSN 0030-8730. http://www.msp.org/pjm/2014/272-1/p04.xhtml. 
  10. ↑ Laubinger, Martin (2006). "Diffeological spaces". Proyecciones 25 (2): 151–178. doi:10.4067/S0716-09172006000200003. ISSN 0717-6279. https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/1542. 
  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. 
  12. ↑ Iglesias-Zemmour, Patrick; Karshon, Yael; Zadka, Moshe (2010). "Orbifolds as diffeologies". Transactions of the American Mathematical Society 362 (6): 2811–2831. doi:10.1090/S0002-9947-10-05006-3. https://www.ams.org/journals/tran/2010-362-06/S0002-9947-10-05006-3/S0002-9947-10-05006-3.pdf. 
  13. ↑ Gürer, Serap; Iglesias-Zemmour, Patrick (2019). "Differential forms on manifolds with boundary and corners" (in en). 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" (in en). Proceedings of the American Mathematical Society 77 (1): 63–67. doi:10.1090/S0002-9939-1979-0539632-8. ISSN 0002-9939. https://www.ams.org/proc/1979-077-01/S0002-9939-1979-0539632-8/. 
  15. ↑ Losik, Mark (1992). "О многообразиях Фреше как диффеологических пространствах" (in ru). Izv. Vyssh. Uchebn. Zaved. Mat. 5: 36–42. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ivm&paperid=4812&option_lang=eng. 
  16. ↑ Losik, Mark (1994). "Categorical differential geometry". Cahiers de Topologie et Géométrie Différentielle Catégoriques 35 (4): 274–290. http://www.numdam.org/item/CTGDC_1994__35_4_274_0/. 
  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" (in fr). C. R. Acad. Sci. Paris Sér. I 301 (4): 127–130. 

External links

  • 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



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