Inmathematics, adiffeology on a set generalizes the concept of smooth charts in adifferentiable manifold, by declaring what constitutes the "smooth parametrizations" into the set.

The concept was first introduced byJean-Marie Souriau in the 1980s under the nameEspace différentiel^[1][2] and later developed by his studentsPaul Donato^[3] andPatrick Iglesias.^[4][5] A related idea was introduced byKuo-Tsaï Chen (陳國才,Chen Guocai) in the 1970s, using convex sets instead of open sets for the domains of the plots.^[6]

Recall that atopological manifold is atopological 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 differentiableatlas, 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.

Adiffeological space consists of a set together with a collection of maps (called adiffeology) 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.

Adiffeology on a set${\displaystyle X}$ consists of a collection of maps, calledplots or parametrizations, fromopen subsets of${\displaystyle {\mathbb{R}}^n}$ (for all${\displaystyle n\ge 0}$) 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 aneighborhood${\displaystyle V\subset U}$ 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 asmooth 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 adiffeological space.

More abstractly, a diffeological space is a concretesheaf on thesite of open subsets of${\displaystyle {\mathbb{R}}^n}$, for all${\displaystyle n\ge 0}$, and open covers.^[7]

A map between diffeological spaces is calledsmooth if and only if its composite with any plot of the first space is a plot of the second space. It is called adiffeomorphism if it is smooth,bijective, and itsinverse 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 acategory where themorphisms are smooth maps. The category of diffeological spaces is closed under many categorical operations: for instance, it isCartesian closed,complete andcocomplete, and more generally it is aquasitopos.^[7]

Any diffeological space is automatically atopological space with the so-calledD-topology:^[8] thefinaltopology such that all plots arecontinuous (with respect to theeuclidean 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 smoothcurves, 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 automaticallylocally path-connected^[10] and a differentiable map between diffeological spaces is automaticallycontinuous between their D-topologies.^[5]

A Cartan-De Rham calculus can be developed in the framework of diffeologies, as well as a suitable adaptation of the notions offiber bundles,homotopy, etc.^[5] However, there is not a canonical definition oftangent spaces andtangent bundles for diffeological spaces.^[11]

• Any set can be endowed with thecoarse (or trivial, or indiscrete)diffeology, i.e. the largest possible diffeology (any map is a plot). The corresponding D-topology is thetrivial topology.
• Any set can be endowed with thediscrete (or fine)diffeology, i.e. the smallest possible diffeology (the only plots are the locally constant maps). The corresponding D-topology is thediscrete topology.
• Any topological space can be endowed with thecontinuous diffeology, whose plots are allcontinuous maps.

Anydifferentiable 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 afull 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${\displaystyle {\mathbb{R}}^n/\mathrm{\Gamma }}$, for${\displaystyle \mathrm{\Gamma }}$ is a finite linear subgroup, and smooth maps between them.^[12]
• Manifolds with boundary or corners, which are modeled onorthants, 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]

• If a set${\displaystyle X}$ is given two different diffeologies, theirintersection is a diffeology on${\displaystyle X}$, called theintersection 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 asubset of the diffeological space${\displaystyle X}$, then thesubspace 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 thesubspace 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 theproduct diffeology on theCartesian 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 theproduct topology of the D-topologies of${\displaystyle X}$ and${\displaystyle Y}$; it is equal to the product topology when${\displaystyle X}$ or${\displaystyle Y}$ islocally compact, but may be finer in general.^[9]
• If${\displaystyle X}$ is a diffeological space and${\displaystyle \sim }$ is anequivalence relation on${\displaystyle X}$, then thequotient diffeology on thequotient 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 thequotient topology of the D-topology of${\displaystyle X}$ (note that this topology may be trivial without the diffeology being trivial).
• Thepushforward 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${\displaystyle X\to X/\sim }$.
• Thepullback 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.
• Thefunctional diffeology between two diffeological spaces${\displaystyle X,Y}$ is the diffeology on the set${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(X,Y)}$ of differentiable maps, whose plots are the maps${\displaystyle \varphi :U\to {\mathcal{C}}^{\mathrm{\infty }}(X,Y)}$ such that${\displaystyle (u,x)\mapsto \varphi (u)(x)}$ 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${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(X,Y)}$ is the smallestlocally path-connected topology containing theweak topology.^[9]

Thewire diffeology (orspaghetti 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${\displaystyle \mathrm{i}\mathrm{d}:{\mathbb{R}}^2\to {\mathbb{R}}^2}$ 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 therank-${\displaystyle r}$-restricted diffeology on a smooth manifold${\displaystyle M}$: a map${\displaystyle U\to M}$ is a plot if and only if the rank of itsdifferential is less or equal than${\displaystyle r}$. For${\displaystyle r=1}$ one recovers the wire diffeology.^[17]

• Quotients gives an easy way to construct non-manifold diffeologies. For example, the set ofreal numbers${\displaystyle \mathbb{R}}$ is a smooth manifold. The quotient${\displaystyle \mathbb{R}/(\mathbb{Z}+\alpha \mathbb{Z})}$, for someirrational${\displaystyle \alpha }$, calledirrational torus, is a diffeological space diffeomorphic to the quotient of the regular2-torus${\displaystyle {\mathbb{R}}^2/{\mathbb{Z}}^2}$ by a line ofslope${\displaystyle \alpha }$. It has a non-trivial diffeology, but its D-topology is thetrivial topology.^[18]
• Combining the subspace diffeology and the functional diffeology, one can define diffeologies on the space of sections of afibre bundle, or the space of bisections of aLie groupoid, etc.

Analogously to the notions ofsubmersions andimmersions between manifolds, there are two special classes of morphisms between diffeological spaces. Asubduction 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, aninduction 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 surjectivesubmersion${\displaystyle f:X\to Y}$ 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 injectiveimmersion need not be an induction. One example is the parametrization of the "figure-eight,"${\displaystyle f:\left(-\frac{\pi }{2},\frac{3\pi }{2}\right)\to {\mathbb{R}}^{\mathbb{2}}}$ 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 strongepimorphisms, and inductions are precisely the strongmonomorphisms. A map that is both a subduction and induction is a diffeomorphism.^[17]

• 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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