Dipheology

In mathematics, dipheology (first invented by Souriau in the 1980s, and later refined by many people) is a generalization of (indefinitely) differentiable manifolds into a more stable category.

If X is a set, a diffeology in X is a set of functions (called plates) from an open subset of some Euclidean space to X such that it is worth the following:

• All constant function is a plate
• For a given function, if every point in the domain has such an environment that the restriction of the function to this environment is a plate, then the function itself is a plate.
• Yeah. p It's a badge, and f is a differentiable function (indefinitely) from an open set of a certain euclide space in the domain of p, then the composition p° f It's a badge.

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

A set together with a diffeology is called a diffeology space.

A function between diffeological spaces is called differentiable, if and only if composed with any plate in the first space gives a plate in the second space.

It is a diffeomorphism if it is differentiable, bijective, and its inverse is also differentiable.

Diffeological spaces, together with differentiable functions such as morphisms, form a category. The isomorphisms in this category are precisely the diffeomorphisms as defined above.

A dipheological space has the D-topology: the finest topology such that all plates are continuous. The open ones in this topology are called D-open ones and are specified as the subsets A of X such that the pre-images under any plate p^-1(A) are open in the usual topology of Rⁿ.

If Y is a subset of the dipheological space X, then Y is itself a dipheological space in a natural way: the plates of Y are those plates of X that have images that are subsets of Y.

Each manifold C^∞ has a dipheology: that in which the plates are differentiable functions from the open subsets of Euclidean spaces to the manifold. In particular, every open subset of Rⁿ has a diffeology.

The manifolds C^∞ together with the (indefinitely) differentiable functions can then be considered as a complete subcategory of the category of dipheological spaces. A diffeomorphic space where every point has a neighborhood from the diffeomorphic D-topology to an open subset of Rⁿ (where n is fixed) is the same as the diffeology generated above for a variety structure.

The notion of a spanning family, due to Patrick Iglesias, is convenient when defining diffeologies: a plate system is a spanning family for a diffology if the diffology is the smallest containing all given plates. In that case, we also say that the diffeology is generated by the given plates.

If X is a dipheological space and ~ is some equivalence relation on X, then the quotient set X/~ has the Dipheology generated by all plate compositions from X with the projection from X to X/~. This is called the quotient dipheology. Note that the quotient D-topology is the D-topology of the quotient diffeology.

This is an easy way to construct dipheologies on non-manifolds. For example, the real numbers R are a dipheological 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 indiscrete topology.