Recall that a topological manifold is a topological space which is locally homeomorphic to ${\displaystyle \mathbb {R} ^{n}}$ . Differentiable 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 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 ${\displaystyle \mathbb {R} ^{n}}$ . Indeed, every smooth manifold has a natural diffeology, consisting of its maximal atlas (all the smooth maps from open subsets of ${\displaystyle \mathbb {R} ^{n}}$  to the manifold). This abstract point of view makes no reference to a specific atlas (and therefore to a fixed dimension ${\displaystyle n}$ ) nor to the underlying topological space, and is therefore suitable to treat examples of objects more general than manifolds.

A diffeology on a set ${\displaystyle X}$  consists of a collection of maps, called plots or parametrizations, from open subsets of ${\displaystyle \mathbb {R} ^{n}}$  (${\displaystyle n\geq 0}$ ) 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 edit

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 edit

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 edit

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]

Trivial examples edit

• 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 edit

• Any differentiable manifold is a diffeological space by considering its maximal atlas (i.e., the plots are all smooth maps from open subsets of ${\displaystyle \mathbb {R} ^{n}}$  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 ${\displaystyle \mathbb {R} ^{n}}$ . For instance, diffeological spaces include orbifolds, which are modeled on quotient spaces ${\displaystyle \mathbb {R} ^{n}/\Gamma }$ , for ${\displaystyle \Gamma }$  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 edit

• 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 the subspace topology of the D-topology of ${\displaystyle X}$ .
• 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 product topology of the D-topologies of ${\displaystyle X}$  and ${\displaystyle Y}$ .
• 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 edit

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 edit

• 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.

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.

When ${\displaystyle X}$  and ${\displaystyle Y}$  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 ${\displaystyle f\circ g}$  is a subduction (respectively, induction) if and only if ${\displaystyle f}$  is a subduction (respectively, ${\displaystyle g}$  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.

• 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