Diffeology

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]

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]

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]

• 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.

• 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]

• 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]

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]

• 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.