Diffeology

In mathematics, diffeology (first invented by Souriau in the 1980s, and later refined by many people) is a generalization of smooth manifolds to a category that is more stable.

If X is a set, a diffeology on X is a set of maps (called plots) from open subsets of some Euclidean space to X such that the following hold:

• Every constant map is a plot.
• For a given map, if every point in the domain has a neighbourhood such that restricting the map to this neighbourhood is a plot, then the map itself is a plot.
• If p is a plot, and f is a smooth (i.e. infinitely often differentiable) function from an open subset of some Euclidean space into the domain of p, then the composition pof is a plot.

Note that the domains of different plots can be subsets of Euclidean spaces of different dimensions.

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 just the diffeomorphisms defined above.

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.

Every smooth (i.e. C^∞) manifold has a diffeology: the one where the plots are the smooth maps from open subsets of Euclidean spaces to the manifold. In particular, every open subset of Rⁿ has a diffeology.

The smooth manifolds with smooth maps can then be seen as a full subcategory of the category of diffeological spaces.

A diffeological space where every point has a D-topology neighbourhood diffeomorphic to an open subset of Rⁿ (where n is fixed) is the same as the diffeology generated as above from a manifold structure.

The notion of a generating family, due to Patrick Iglesias, is convenient in defining diffeologies: a set of plots is a generating family for a diffeology if the diffeology is the smallest diffeology containing all the given plots. In that case, we also say that the diffeology is generated by the given plots.

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. Note that the quotient D-topology is the D-topology of the quotient diffeology.

This is a way to easily get non-manifold diffeologies. For example, the real numbers R are a diffeological space (they are a manifold). R/(Z + αZ), for some irrational α, is the irrational torus. It has a diffeology, but the D-topology for it is the trivial topology.

External link

• Patrick Iglesias-Zemmour: Introduction to Diffeology (a working document)