Diffeological space
In mathematics, a diffeology on a set declares what the smooth parametrizations in the set are. In some sense a diffeology generalizes the concept of smooth charts in a differentiable manifold.
The concept was first introduced by Jean-Marie Souriau in the 1980s and developed first by his students Paul Donato (homogeneous spaces and coverings) and Patrick Iglesias (diffeological fiber bundles, higher homotopy etc.), later by other people. 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.
If X is a set, a diffeology on X is a set of maps, called plots, from open subsets of Rn (n ≥ 0) to X such that the following hold:
Note that the domains of different plots can be subsets of Rn for different values of n.
A set together with a diffeology is called a diffeological space.
A map between diffeological spaces is called differentiable if and only if composing it with every plot of the first space is a plot of the second space. It is a diffeomorphism if it is differentiable, bijective, and its inverse is also differentiable.
The diffeological spaces, together with differentiable maps as morphisms, form a category. The isomorphisms in this category are the diffeomorphisms defined above. The category of diffeological spaces is closed under many categorical operations.
A diffeological space has the D-topology: the finest topology such that all plots are continuous.
If Y is a subset of the diffeological space X, then Y is itself a diffeological space in a natural way: the plots of Y are those plots of X whose images are subsets of Y.
If X is a diffeological space and ~ is some equivalence relation on X, then the quotient set X/~ has the diffeology generated by all compositions of plots of X with the projection from X to X/~. This is called the quotient diffeology. The quotient D-topology is the D-topology of the quotient diffeology, and that this topology may be trivial without the diffeology being trivial.
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