nLab
diffeological space

Diffeological spaces are a kind of generalized smooth spaces, namely concrete smooth spaces.

Diffeological spaces were originally described by J.M. Souriau in 1980. They have subsequently been developed by Patrick Iglesias-Zemmour who is writing a book on the subject.

Definition

Let $\mathrm{Diff}$ denote the site whose objects are open subsets of Euclidean spaces and whose morphisms are $C^{\mathrm{\infty }}$-maps.

A diffeological space is a pair $(X,𝒟)$ where $X$ is a set and $𝒟$ is a sheaf on $\mathrm{Diff}$ that is a subsheaf of the sheaf $U\mapsto X^{\mid U\mid }$.

A morphism of diffeological spaces is a map of the underlying sets which defines a natural transformation on the sheafs.

A diffeological space is an example of a concrete sheaf on a concrete site? and as such is covered by Baez and Hoffnung in 0807.1704.

The concreteness condition on the sheaf is a reiteration of the fact that a diffeological space is a subsheaf of the sheaf $U\mapsto X^{\mid U\mid }$. In this way, one does not have to explicitly mention the underlying set $X$ as it is determined by the sheaf on the one-point open subset of ${ℝ}^0$.

References

• Groupes diff'erentiels

• Patrick Iglesias-Zemmour, Diffeology (web, pdf)

The thesis

• Patrick Iglesias-Zemmour, Fibrations difféologie et Homotopie, PhD thesis pdf

contains some useful material that hasn’t yet made it into the book.

• John C. Baez, Alexander E. Hoffnung, Convenient Categories of Smooth Spaces (arXiv)