Diffeology

Not to be confused with Diffiety.

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

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 ${\displaystyle \mathbb {R} ^{n}}$. Differentiable manifolds (also called smooth manifolds) generalize the notion of smoothness on ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle \mathbb {R} ^{n}}$ to the manifold which are used to "pull back" the differential structure from ${\displaystyle \mathbb {R} ^{n}}$ to the manifold.

A diffeological space consists of a set together with a collection of maps (called a diffeology) satisfying suitable axioms, which are used to characterize smoothness of the space in a way similar to charts of an atlas.

A smooth manifold can be equivalently defined as a diffeological space which is locally diffeomorphic to ${\displaystyle \mathbb {R} ^{n}}$. But there are many diffeological spaces which do not carry any local model, nor a sufficiently interesting underlying topological space. Diffeology is therefore suitable to treat examples of objects more general than manifolds.

Formal definition

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

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

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

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 ${\displaystyle \mathbb {R} ^{n}}$).

In other words, a subset ${\displaystyle U\subset X}$ is open if and only if ${\displaystyle f^{-1}(U)}$ is open for any plot ${\displaystyle f}$ on ${\displaystyle X}$. Actually, the D-topology is completely determined by smooth curves, i.e. a subset ${\displaystyle U\subset X}$ is open if and only if ${\displaystyle c^{-1}(U)}$ is open for any smooth map ${\displaystyle c:\mathbb {R} \to X}$.^[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 ${\displaystyle M}$ can be assigned the diffeology consisting of all smooth maps from all open subsets of Euclidean spaces into it. This diffeology will contain not only the charts of ${\displaystyle M}$, but also all smooth curves into ${\displaystyle M}$, all constant maps (with domains open subsets of Euclidean spaces), etc. The D-topology recovers the original manifold topology. With this diffeology, a map between two smooth manifolds is smooth in the usual sense if and only if it is smooth in the diffeological sense. Accordingly, smooth manifolds with smooth maps form a full subcategory of the category of diffeological spaces.

This procedure similarly assigns diffeologies to other spaces that possess a smooth structure that is determined by a local model. More precisely, each of the examples below form a full subcategory of diffeological spaces.

• Orbifolds, which are modeled on quotient spaces ${\displaystyle \mathbb {R} ^{n}/\Gamma }$, for ${\displaystyle \Gamma }$ is a finite linear subgroup, and smooth maps between them.^[12]
• Manifolds with boundary or corners, which are modeled on orthants, and smooth maps between them.^[13]
• Banach manifolds, which are modeled on Banach spaces, and smooth maps between them.^[14]
• Fréchet manifolds, which are modeled on Fréchet spaces, and smooth maps between them.^[15][16]

Constructions from other diffeological spaces

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

Wire/spaghetti diffeology

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

This example can be enlarged to diffeologies whose plots factor locally through ${\displaystyle \mathbb {R} ^{r}}$. More generally, one can consider the rank-${\displaystyle r}$-restricted diffeology on a smooth manifold ${\displaystyle M}$: a map ${\displaystyle U\to M}$ is a plot if and only if the rank of its differential is less or equal than ${\displaystyle r}$. For ${\displaystyle r=1}$ 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 ${\displaystyle \mathbb {R} }$ is a smooth manifold. The quotient ${\displaystyle \mathbb {R} /(\mathbb {Z} +\alpha \mathbb {Z} )}$, for some irrational ${\displaystyle \alpha }$, called irrational torus, is a diffeological space diffeomorphic to the quotient of the regular 2-torus ${\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ by a line of slope ${\displaystyle \alpha }$. 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 ${\displaystyle f:X\to Y}$ between diffeological spaces such that the diffeology of ${\displaystyle Y}$ is the pushforward of the diffeology of ${\displaystyle X}$. Similarly, an induction is an injective function ${\displaystyle f:X\to Y}$ between diffeological spaces such that the diffeology of ${\displaystyle X}$ is the pullback of the diffeology of ${\displaystyle Y}$. Note that subductions and inductions are automatically smooth.

It is instructive to consider the case where ${\displaystyle X}$ and ${\displaystyle Y}$ are smooth manifolds.

• Every surjective submersion ${\displaystyle f:X\to Y}$ is a subduction.
• A subduction need not be a surjective submersion. One example is ${\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }$ given by ${\displaystyle f(x,y):=xy}$.
• An injective immersion need not be an induction. One example is the parametrization of the "figure-eight," ${\displaystyle f:\left(-{\frac {\pi }{2}},{\frac {3\pi }{2}}\right)\to \mathbb {R^{2}} }$ given by ${\displaystyle f(t):=(2\cos(t),\sin(2t))}$.
• An induction need not be an injective immersion. One example is the "semi-cubic," ${\displaystyle f:\mathbb {R} \to \mathbb {R} ^{2}}$given by ${\displaystyle f(t):=(t^{2},t^{3})}$.^[19][20]

In the category of diffeological spaces, subductions are precisely the strong epimorphisms, and inductions are precisely the strong monomorphisms. A map that is both a subduction and induction is a diffeomorphism.^[17]

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. 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. 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. 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.
19. Karshon, Yael; Miyamoto, David; Watts, Jordan (2022-04-21). "Diffeological submanifolds and their friends". arXiv:2204.10381 [math.DG].
20. Joris, Henri (1982-09-01). "Une C∞-application non-immersive qui possède la propriété universelle des immersions". Archiv der Mathematik (in French). 39 (3): 269–277. doi:10.1007/BF01899535. ISSN 1420-8938.

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