Diffeology

In mathematics, a diffeology generalizes the concept of smooth maps, which can be naturally defined for vector spaces. The corresponding category is strongly stable under many categorical 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.

Definition

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:

• Every constant map is a plot.
• For a given map, if every point in the domain has a neighborhood such that restricting the map to this neighborhood 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 ∘ f is a plot.

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 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.

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.

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.

Smooth manifolds

Differentiable manifolds also generalize smoothness. They are normally defined as topological manifolds with an atlas, whose transition maps are smooth, which is used to pull back the differential structure.

Every smooth manifold defined in this way has a natural diffeology, for which the plots correspond to the smooth maps from open subsets of Rⁿ to the manifold. With this diffeology, a map between two smooth manifolds is smooth if and only if it is differentiable in the diffeological sense. Hence the smooth manifolds with smooth maps form a full subcategory of the diffeological spaces.

This allows one to give an alternative definition of smooth manifold which makes no reference to transition maps or to a specific atlas: a smooth manifold is a diffeological space which is locally diffeomorphic to Rⁿ.

The relationship between smooth manifolds and diffeological spaces is analogous to the relationship between topological manifolds and topological spaces.

Examples

• Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.
• Any smooth manifold is a diffeological space.
• Any quotient of a diffeological space is a diffeological space. 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 links

• Patrick Iglesias-Zemmour: Diffeology (a working document)
• Patrick Iglesias-Zemmour: Diffeology (many documents)