Diffeological Spaces: Examples and Relations to Smooth Spaces
Diffeology provides a flexible framework for studying smooth structures on spaces that may not fit the rigid definition of a traditional manifold. By focusing on plots—maps from open subsets of Euclidean space into a set—diffeology allows mathematicians to define smoothness on a vast array of objects, from simple sets to complex infinite-dimensional spaces.
Fundamental Examples of Diffeologies
Every set can be equipped with at least two basic types of diffeologies, representing the two extremes of smooth structure:
- Coarse (Trivial or Indiscrete) Diffeology: This is the largest possible diffeology, where every map into the set is considered a plot. Its associated D-topology (the topology induced by the diffeology) is the trivial topology.
- Discrete (Fine) Diffeology: This is the smallest possible diffeology, consisting only of locally constant maps. Its associated D-topology is the discrete topology.
Beyond these extremes, any topological space can be given a continuous diffeology, where the plots are defined as the continuous maps.
Diffeologies on Euclidean Space $\mathbb{R}^n$
Euclidean space $\mathbb{R}^n$ serves as a primary example of how different diffeologies can exist on the same underlying set:
- Standard Diffeology: Plots are maps $p: U \to \mathbb{R}^n$ that are smooth in the traditional sense of multivariable calculus.
- Wire (Spaghetti) Diffeology: A map is a plot if it factors locally through $\mathbb{R}$. Specifically, for every point in the domain, there exists a neighborhood where the map can be written as the composition of two smooth functions: one mapping to $\mathbb{R}$ and another mapping from $\mathbb{R}$ to $\mathbb{R}^n$. For $n \geq 2$, this differs from the standard diffeology; for example, the identity map on $\mathbb{R}^n$ is not a plot in the wire diffeology.
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Rank-Restricted Diffeologies
The concept of the wire diffeology can be generalized to the rank-r restricted diffeology on a smooth manifold $M$. In this case, a map is a plot if it is smooth and the rank of its differential is less than or equal to $r$. When $r=1$, this recovers the wire diffeology.
Key Facts
- Diffeological spaces generalize manifolds, orbifolds, and Fréchet manifolds.
- The D-topology is the topology naturally induced by a given diffeology.
- Standard smooth manifolds fully embed into the category of diffeological spaces ($\mathsf{Dflg}$).
- The wire diffeology is a specific case of a rank-restricted diffeology where $r=1$.
- Diffeological orbifolds provide a standardized notion of smooth maps, which is often lacking in classical orbifold theory.
Relation to Other Smooth Spaces
Diffeological spaces act as a broad umbrella for various mathematical objects. To formalize this, $\mathsf{Dflg}$ is viewed as a concrete category over the category of topological spaces $\mathsf{Top}$ via the D-topology functor. A diffeological space is said to be locally modeled by a collection of spaces if every point has a D-open neighborhood diffeomorphic to a D-open subset of a space in that collection.
| Space Type | Local Model Collection | D-Topology Requirement |
|---|---|---|
| Smooth Manifolds | $\{y(\mathbb{R}^n)\}$ | Hausdorff and second-countable |
| Manifolds with Boundary | $\{y(O) \mid O \text{ is a half-space}\}$ | Hausdorff and second-countable |
| Manifolds with Corners | $\{y(O) \mid O \text{ is an orthant}\}$ | Hausdorff and second-countable |
| Fréchet Manifolds | $\{y(E) \mid E \text{ is a Fréchet space}\}$ | Hausdorff |
| Orbifolds | $\{\mathbb{R}^n / \Gamma\}$ | Hausdorff |
Manifolds and Infinite-Dimensional Spaces
Finite-dimensional smooth manifolds, including those with boundaries or corners, fully embed into $\mathsf{Dflg}$. For these spaces, a diffeologically smooth map is identical to a smooth map in the traditional sense, and the D-topology matches the original manifold topology.
This embedding extends to infinite-dimensional spaces. Fréchet manifolds and Banach manifolds (the latter proven by Hain, the former by Losik) also fully embed into $\mathsf{Dflg}$. Similarly, manifolds modeled on convenient vector spaces are compatible with this framework.
Orbifolds
A classical orbifold is locally modeled by quotients $\mathbb{R}^n / \Gamma$, where $\Gamma$ is a finite subgroup of linear transformations. Since these quotients are naturally diffeological spaces, they generate a unique diffeology on the orbifold. A diffeological orbifold is a diffeological space locally modeled by these quotients with a Hausdorff D-topology.
One significant advantage of the diffeological approach is that it automatically provides a consistent definition of smooth maps between orbifolds, a concept that is not standardized in classical orbifold theory.
Frequently Asked Questions
What is the difference between the coarse and discrete diffeologies?
The coarse diffeology is the largest possible, treating every map into the set as a plot, resulting in a trivial topology. The discrete diffeology is the smallest, treating only locally constant maps as plots, resulting in a discrete topology.
How does the wire diffeology differ from the standard diffeology on $\mathbb{R}^n$?
In the standard diffeology, any smooth map is a plot. In the wire diffeology, plots must locally factor through $\mathbb{R}$. For $n \geq 2$, the identity map on $\mathbb{R}^n$ is a plot in the standard diffeology but not in the wire diffeology.
Can all smooth manifolds be treated as diffeological spaces?
Yes, the category of finite-dimensional smooth manifolds fully embeds into the category of diffeological spaces. In this embedding, the D-topology is the same as the original manifold topology, and smooth maps remain the same.
What is a rank-r restricted diffeology?
It is a diffeology on a smooth manifold where a map is considered a plot if it is smooth and the rank of its differential is less than or equal to $r$.
Why are diffeological orbifolds useful compared to classical orbifolds?
Diffeological orbifolds provide a standardized and natural definition of smooth maps between them, whereas the notion of a smooth map between classical orbifolds is not universally standardized.