Diffeology
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In mathematics, a diffeology is a generalization of smooth manifolds to a category that is more stable by the main set operations. The concept was first introduced by Kuo Tsaï Chen in the 1970s, under the name "differential spaces", then rediscovered by Souriau in the 1980s and later refined by many people.
If X is a set, a diffeology on X is a set of maps (called plots) from open subsets of any real vector spaces 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 function from an open subset of some real vector space into the domain of p, then the composition p o f is a plot.
Note that the domains of different plots can be subsets of real vector 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 natural diffeology: the one where the plots are the smooth maps from open subsets of real vector spaces to the manifold. In particular, every open subset of Rn has a smooth 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 an easy way to construct non-manifold diffeologies. For example, the set of real numbers R is a smooth manifold. The quotient R/(Z + αZ), for some irrational α, is the irrational torus. A diffeological space diffeomorphic to the quotient of the regular 2-torus R2/Z2 by a line of slope α. It has a non trivial diffeology, but its D-topology is the trivial topology.
External link
- Patrick Iglesias-Zemmour: Introduction to Diffeology (a working document) (http://www.umpa.ens-lyon.fr/~iglesias/articles/Diffeology/Diffeology.pdf)
- Patrick Iglesias-Zemmour: Diffeology (many documents) (http://www.umpa.ens-lyon.fr/~iglesias/)