Encyclopedia > Diffeology

  Article Content


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 different Euclidean spaces.

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Anna Karenina

... the wife of the bureaucrat Karenin. She is invited from her home in St Petersburg to Moscow to mediate the marital discord of her brother Stepan (Stiva) Oblonsky and his ...

This page was created in 34.9 ms