Inmathematics, adiffeology on a set generalizes the concept of a smooth atlas of adifferentiable manifold, by declaring only what constitutes the "smooth parametrizations" into the set. A diffeological space is a set equipped with a diffeology. Many of the standard tools ofdifferential geometry extend to diffeological spaces, which beyond manifolds include arbitrary quotients of manifolds, arbitrary subsets of manifolds, and spaces of mappings between manifolds.

Thedifferential calculus on${\displaystyle {\mathbb{R}}^n}$, or, more generally, on finite dimensionalvector spaces, is one of the most impactful successes of modern mathematics. Fundamental to its basic definitions and theorems is the linear structure of the underlying space.^[1][2]

The field ofdifferential geometry establishes and studies the extension of the classical differential calculus to non-linear spaces. This extension is made possible by the definition of asmooth manifold, which is also the starting point for diffeological spaces.

A smooth${\displaystyle n}$-dimensional manifold is a set${\displaystyle M}$ equipped with a maximalsmooth atlas, which consists of injective functions, calledcharts, of the form${\displaystyle \varphi :U\to M}$, where${\displaystyle U}$ is an open subset of${\displaystyle {\mathbb{R}}^n}$, satisfying some mutual-compatibility relations. The charts of a manifold perform two distinct functions, which are often syncretized:^[3][4][5]

• They dictate the local structure of the manifold. The chart${\displaystyle \varphi :U\to M}$ identifies its image in${\displaystyle M}$ with its domain${\displaystyle U}$. This is convenient because the latter is simply an open subset of aEuclidean space.
• They define the class of smooth maps between manifolds. These are the maps to which the differential calculus extends. In particular, the charts determine smooth functions (smooth maps${\displaystyle M\to \mathbb{R}}$), smoothcurves (smooth maps${\displaystyle \mathbb{R}\to M}$), smoothhomotopies (smooth maps${\displaystyle {\mathbb{R}}^2\to M}$), etc.

A diffeology generalizes the structure of a smooth manifold by abandoning the first requirement for an atlas, namely that the charts give a local model of the space, while retaining the ability to discuss smooth maps into the space.^[6][7][8]

Adiffeological space is a set${\displaystyle X}$ equipped with adiffeology: a collection of maps$${\displaystyle \{p:U\to X\mid U\text{ is an open subset of }{\mathbb{R}}^n,\text{ and }n\ge 0\},}$$whose members are calledplots, that satisfies some axioms. The plots are not required to be injective, and can (indeed, must) have as domains the open subsets of arbitrary Euclidean spaces.

A smooth manifold can be viewed as a diffeological space which is locally diffeomorphic to${\displaystyle {\mathbb{R}}^n}$. In general, while not giving local models for the space, the axioms of a diffeology still ensure that the plots induce a coherent notion of smooth functions, smooth curves, smooth homotopies, etc. Diffeology is therefore suitable to treat objects more general than manifolds.^[6][7][8]

Let${\displaystyle M}$ and${\displaystyle N}$ be smooth manifolds. A smooth homotopy of maps${\displaystyle M\to N}$ is a smooth map${\displaystyle H:\mathbb{R}\times M\to N}$. For each${\displaystyle t\in \mathbb{R}}$, the map${\displaystyle H_t:=H(t,\cdot ):M\to N}$ is smooth, and the intuition behind a smooth homotopy is that it is a smooth curve into the space of smooth functions${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(M,N)}$ connecting, say,${\displaystyle H_0}$ and${\displaystyle H_1}$. But${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(M,N)}$ is not a finite-dimensional smooth manifold, so formally we cannot yet speak of smooth curves into it.

On the other hand, the collection of maps$${\displaystyle \{p:U\to {\mathcal{C}}^{\mathrm{\infty }}(M,N)\mid \text{ the map }U\times M\to N,(r,x)\mapsto p(r)(x)\text{ is smooth}\}}$$is a diffeology on${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(M,N)}$. With this structure, the smooth curves (a notion which is now rigorously defined) correspond precisely to the smooth homotopies.^[6][7][8]

The concept of diffeology was first introduced byJean-Marie Souriau in the 1980s under the nameespace différentiel.^[9][10] Souriau's motivating application for diffeology was to uniformly handle the infinite-dimensional groups arising from his work ingeometric quantization. Thus the notion of diffeological group preceded the more general concept of a diffeological space. Souriau's diffeological program was taken up by his students, particularlyPaul Donato^[11] andPatrick Iglesias-Zemmour,^[12] who completed early pioneering work in the field.

A structure similar to diffeology was introduced byKuo-Tsaï Chen (陳國才,Chen Guocai) in the 1970s, in order to formalize certain computations with path integrals. Chen's definition usedconvex sets instead of open sets for the domains of the plots.^[13] The similarity between diffeological and "Chen" structures can be made precise by viewing both as concrete sheaves over the appropriate concrete site.^[14]

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${\displaystyle p:U\to X}$, if every point in${\displaystyle U}$ has aneighborhood${\displaystyle V\subset U}$ such that${\displaystyle p{|}_V}$ is a plot, then${\displaystyle p}$ 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${\displaystyle n\ge 0}$, andopen covers.^[14]

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. Equipping the open subsets of Euclidean spaces with their standard diffeology (as defined in the next section), the plots into a diffeological space${\displaystyle X}$ are precisely the smooth maps from${\displaystyle U}$ to${\displaystyle X}$.

Diffeological spaces constitute the objects of acategory, denoted by${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$, whosemorphisms are smooth maps. The category${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$ is closed under many categorical operations: for instance, it isCartesian closed,complete andcocomplete, and more generally it is aquasitopos.^[14]

Any diffeological space is atopological space when equipped with theD-topology:^[12] thefinal topology 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${\displaystyle p^{-1}(U)}$ is open for any plot${\displaystyle p}$ 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${\displaystyle c:\mathbb{R}\to X}$.^[15] The D-topology is automaticallylocally path-connected^[16]

A smooth map between diffeological spaces is automaticallycontinuous between their D-topologies.^[6] Therefore we have the functor${\displaystyle D:\mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}\to \mathsf{T}\mathsf{o}\mathsf{p}}$, from the category of diffeological spaces to the category of topological spaces, which assigns to a diffeological space its D-topology. This functor realizes${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$ as aconcrete category over${\displaystyle \mathsf{T}\mathsf{o}\mathsf{p}}$.

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.^[6] However, there is not a canonical definition oftangent spaces andtangent bundles for diffeological spaces.^[17]

Any set carries at least two diffeologies:

• thecoarse (or trivial, or indiscrete) diffeology, consisting of every map into the set. This is the largest possible diffeology. The corresponding D-topology is thetrivial topology.
• thediscrete (or fine) diffeology, consisting of the locally constant maps into the set. This is the smallest possible diffeology. The corresponding D-topology is thediscrete topology.

Any topological space can be endowed with thecontinuous diffeology, whose plots are thecontinuous maps.

The Euclidean space${\displaystyle {\mathbb{R}}^n}$admits several diffeologies beyond those listed above.

• Thestandard diffeology on${\displaystyle {\mathbb{R}}^n}$ consists of those maps${\displaystyle p:U\to {\mathbb{R}}^n}$ which are smooth in the usual sense of multivariable calculus.
• Thewire (or spaghetti) diffeology on${\displaystyle {\mathbb{R}}^n}$ is the diffeology whose plots factor locally through${\displaystyle \mathbb{R}}$. More precisely, a map${\displaystyle p:U\to {\mathbb{R}}^n}$ 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 smooth functions${\displaystyle F:V\to \mathbb{R}}$ and${\displaystyle q:\mathbb{R}\to {\mathbb{R}}^n}$. This diffeology does not coincide with the standard diffeology on${\displaystyle {\mathbb{R}}^n}$ when${\displaystyle n\ge 2}$: for instance, the identity${\mathbb{R}}^n\to X={\mathbb{R}}^n$ is not a plot for the wire diffeology.^[6]
• The previous example can be enlarged to diffeologies whose plots factor locally through${\displaystyle {\mathbb{R}}^r}$, yielding therank-${\displaystyle r}$-restricted diffeology on a smooth manifold${\displaystyle M}$: a map${\displaystyle U\to M}$ is a plot if and only if it is smooth and the rank of itsdifferential is less than or equal than${\displaystyle r}$. For${\displaystyle r=1}$ one recovers the wire diffeology.^[18]

Diffeological spaces generalize manifolds, but they are far from the only mathematical objects to do so. For instance manifolds with corners, orbifolds, and infinite-dimensional Fréchet manifolds are all well-established alternatives. This subsection makes precise the extent to which these spaces are diffeological.

We view${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$ as a concrete category over the category of topological spaces${\displaystyle \mathsf{T}\mathsf{o}\mathsf{p}}$ via the D-topology functor${\displaystyle D:\mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}\to \mathsf{T}\mathsf{o}\mathsf{p}}$. If${\displaystyle U:\mathsf{C}\to \mathsf{T}\mathsf{o}\mathsf{p}}$ is another concrete category over${\displaystyle \mathsf{T}\mathsf{o}\mathsf{p}}$, we say that a functor${\displaystyle E:\mathsf{C}\to \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$ is an embedding (of concrete categories) if it is injective on objects and faithful, and${\displaystyle D\circ E=U}$. To specify an embedding, we need only describe it on objects; it is necessarily the identity map on arrows.

We will say that a diffeological space${\displaystyle X}$ islocally modeled by a collection of diffeological spaces${\displaystyle \mathcal{E}}$ if around every point${\displaystyle x\in X}$, there is a D-open neighbourhood${\displaystyle U}$, a D-open subset${\displaystyle V}$ of some${\displaystyle E\in \mathcal{E}}$, and a diffeological diffeomorphism${\displaystyle U\to V}$.^[6][19]

The category of finite-dimensional smooth manifolds (allowing those with connected components of different dimensions) fully embeds into${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$. The embedding${\displaystyle y}$ assigns to a smooth manifold${\displaystyle M}$ the canonical diffeology$${\displaystyle \{p:U\to M\mid p\text{ is smooth in the usual sense}\}.}$$In particular, a diffeologically smooth map between manifolds is smooth in the usual sense, and the D-topology of${\displaystyle y(M)}$ is the original topology of${\displaystyle M}$. Theessential image of this embedding consists of those diffeological spaces that are locally modeled by the collection${\displaystyle \{y({\mathbb{R}}^n)\}}$, and whose D-topology isHausdorff andsecond-countable.^[6]

The category of finite-dimensional smoothmanifolds with boundary (allowing those with connected components of different dimensions) similarly fully embeds into${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$. The embedding is defined identically to the smooth case, except "smooth in the usual sense" refers to the standard definition of smooth maps between manifolds with boundary. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection${\displaystyle \{y(O)\mid O\text{ is a half-space}\}}$, and whose D-topology is Hausdorff and second-countable. The same can be done in more generality formanifolds with corners, using the collection${\displaystyle \{y(O)\mid O\text{ is an orthant}\}}$.^[20]

The category ofFréchet manifolds similarly fully embeds into${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$. Once again, the embedding is defined identically to the smooth case, except "smooth in the usual sense" refers to the standard definition of smooth maps between Fréchet spaces. The essential image of this embedding consists of those diffeological spaces that are locally modeled by the collection${\displaystyle \{y(E)\mid E\text{ is a Fréchet space}\}}$, and whose D-topology is Hausdorff.

The embedding restricts to one of the category ofBanach manifolds. Historically, the case of Banach manifolds was proved first, by Hain,^[21] and the case of Fréchet manifolds was treated later, by Losik.^[22][23] The category of manifolds modeled onconvenient vector spaces also similarly embeds into${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$.^[24][25]

A (classical)orbifold${\displaystyle X}$ is a space that is locally modeled by quotients of the form${\displaystyle {\mathbb{R}}^n/\mathrm{\Gamma }}$, where${\displaystyle \mathrm{\Gamma }}$ is afinite subgroup of linear transformations. On the other hand, each model${\displaystyle {\mathbb{R}}^n/\mathrm{\Gamma }}$ is naturally a diffeological space (with the quotient diffeology discussed below), and therefore the orbifold charts generate a diffeology on${\displaystyle X}$. This diffeology is uniquely determined by the orbifold structure of${\displaystyle X}$.

Conversely, a diffeological space that is locally modeled by the collection${\displaystyle \{{\mathbb{R}}^n/\mathrm{\Gamma }\}}$ (and with Hausdorff D-topology) carries a classical orbifold structure that induces the original diffeology, wherein the local diffeomorphisms are the orbifold charts. Such a space is called a diffeological orbifold.^[26]

Whereas diffeological orbifolds automatically have a notion of smooth map between them (namely diffeologically smooth maps in${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$), the notion of a smooth map between classical orbifolds is not standardized.

If orbifolds are viewed asdifferentiable stacks presented by étale properLie groupoids, then there is a functor from the underlying 1-category of orbifolds, and equivalent maps-of-stacks between them, to${\displaystyle \mathsf{D}\mathsf{f}\mathsf{l}\mathsf{g}}$. Its essential image consists of diffeological orbifolds, but the functor is neither faithful nor full.^[27]

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 finer than the intersection of the D-topologies of the original diffeologies.

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${\displaystyle Y}$. Precisely, a map${\displaystyle p:U\to X\times Y}$ necessarily has the form${\displaystyle p(u)=(x(u),y(u))}$ for maps${\displaystyle x:U\to X}$ and${\displaystyle y:U\to Y}$. The map${\displaystyle p}$ is a plot in the product diffeology if and only if${\displaystyle x}$ and${\displaystyle y}$ are plots of${\displaystyle X}$ and${\displaystyle Y}$, respectively. This generalizes to products of arbitrary collections of spaces.

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.^[15]

Given a map${\displaystyle f:X\to Y}$ from a set${\displaystyle X}$ to a diffeological space${\displaystyle Y}$, thepullback diffeology on${\displaystyle X}$ consists of those maps${\displaystyle p:U\to X}$ 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${\displaystyle f}$ smooth.

If${\displaystyle X}$ is asubset of the diffeological space${\displaystyle Y}$, then thesubspace diffeology on${\displaystyle X}$ is the pullback diffeology induced by the inclusion${\displaystyle X\hookrightarrow Y}$. In this case, the D-topology of${\displaystyle X}$ is equal to thesubspace topology of the D-topology of${\displaystyle Y}$ if${\displaystyle Y}$ is open, but may be finer in general.

Given a map${\displaystyle f:X\to Y}$ from diffeological space${\displaystyle X}$ to a set${\displaystyle Y}$, thepushforward diffeology on${\displaystyle Y}$ is the diffeology generated by the compositions${\displaystyle f\circ p}$, for plots${\displaystyle p:U\to X}$ of${\displaystyle X}$. In other words, the pushforward diffeology is the smallest diffeology on${\displaystyle Y}$ making${\displaystyle f}$ smooth.

If${\displaystyle X}$ is a diffeological space and${\displaystyle \sim }$ is anequivalence relation on${\displaystyle X}$, then thequotient diffeology on thequotient set${\displaystyle X/\sim }$ is the pushforward diffeology induced by the quotient map${\displaystyle X\to X/\sim }$. 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.

Quotients often give rise to 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${\displaystyle \alpha }$, called theirrational 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, although its D-topology is thetrivial topology.^[28]

Thefunctional diffeology on the set${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(X,Y)}$ of smooth maps between two diffeological spaces${\displaystyle X}$ and${\displaystyle Y}$ is the diffeology whose plots are the maps${\displaystyle \varphi :U\to {\mathcal{C}}^{\mathrm{\infty }}(X,Y)}$ such that$${\displaystyle U\times X\to Y,(u,x)\mapsto \varphi (u)(x)}$$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 the Whitney${\displaystyle C^{\mathrm{\infty }}}$ topology.^[15]

Taking the subspace diffeology of a functional diffeology, one can define diffeologies on the space ofsections of afibre bundle, or the space of bisections of aLie groupoid, etc.

If${\displaystyle M}$ is a compact smooth manifold, and${\displaystyle F\to M}$ is a smooth fiber bundle over${\displaystyle M}$, then the space of smooth sections${\displaystyle \mathrm{\Gamma }(F)}$ of the bundle is frequently equipped with the structure of a Fréchet manifold.^[29] Upon embedding this Fréchet manifold into the category of diffeological spaces, the resulting diffeology coincides with the subspace diffeology that${\displaystyle \mathrm{\Gamma }(F)}$ inherits from the functional diffeology on${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(M,F)}$.^[30]

Analogous 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${\displaystyle Y}$. 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$${\displaystyle f:{\mathbb{R}}^2\to \mathbb{R},f(x,y):=xy.}$$
• An injectiveimmersion need not be an induction. One example is the parametrization of the "figure-eight,"

$${\displaystyle f:\left(-\frac{\pi }{2},\frac{3\pi }{2}\right)\to {\mathbb{R}}^{\mathbb{2}},f(t):=(2\cos (t),\sin (2t)).}$$

• An induction need not be an injective immersion. One example is the "semi-cubic,"^[31][32]

$${\displaystyle f:\mathbb{R}\to {\mathbb{R}}^2,f(t):=(t^2,t^3).}$$

In the category of diffeological spaces, subductions are precisely the strongepimorphisms, and inductions are precisely the strongmonomorphisms.^[18] A map that is both a subduction and induction is a diffeomorphism.

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