Diffeological Space

General Idea

There is a lemma (see BIKW17, Sec 1.)

Lemma(Smooth maps) . A map $f: U\to V$ between open subsets $U\subset \mathbb{R}^m$ and $V\subseteq \mathbb{R}^n$ is smooth iff the following three equivalent conditions are satisfied
  1. (Pullback stability) For each open subset $W\subseteq \mathbb{R}^k$ and smooth $h:W\to U$ the composition $f\circ h:W\to V$ is smooth.
  2. (Existence of "concrete subobjects") For every real-valued smooth function $g\in C^\infty(V,\mathbb{R})$ the composition $g\circ f: U\to \mathbb{R}$ is smooth.
  3. (Pullback stability of "concrete subobjects") For every smooth curve $\gamma:\mathbb{R}\to U$ the composition $f\circ \gamma:\mathbb{R}\to V$ is smooth.
The naming for the conditions should not be taken seriously. Notice the similarities with the definition of Grothendieck topology.

This says that

  1. $C^\infty(W,U)$ (with $W$ ranging over all opensubsets of all the $\mathbb{R}^k$),
  2. $C^\infty(\mathbb{R},U)$,
  3. $C^\infty(U,\mathbb{R})$,

all equivalently determine the smoothness of maps defined on $U\subseteq \mathbb{R}^m$.

The idea now is to define the notion of smoothness with these data just as in the case of topological spaces where one define the notion of continuity by “imposing” openness on some sets:

  1. (Diffeological space) Impose a family $\mathcal{D}_X$ of smooth maps $W\to X$ to $X$. A map $X\to Y$ is smooth iff for all $W\to X$ in the family $\mathcal{D}_X$ the pullback stability holds, i.e. the pullback of $W\to X$ by $X\to Y$, $W\to X \to Y$, is smooth.
  2. (Sikorski space) Impose a family $\mathcal{F}_X$ of smooth functions $X\to \mathbb{R}$. A map $X\to Y$ is smooth iff it is smooth when pulled back by smooth functions, i.e. $X\to Y\to \mathbb{R}$ is smooth for $Y\to \mathbb{R}$ in $\mathcal{F}_Y$.
  3. (Froelicher space) A function $X\to Y$ is a smooth map (thus in the family of smooth maps $\mathcal{F}_X$ originating from $X$) iff for all curves in the family $\mathcal{C}_X$ of smooth curves $\mathbb{R}\to X$ on $X$ and all maps in the family $\mathcal{S}_X$ of smooth maps $\mathbb{R}\to X\to Y$ ($Y$ is variable) the composition $\mathbb{R}\to X\to Y$ is in $\mathcal{C}_Y$. The imposed family are $\mathcal{S}_X$ and $\mathcal{C}_X$.

The ideas for the spaces should, of course, be supplemented with some consistency conditions.

Hence equipping a space $X$ with a smooth structure is transfering the syntax of smoothness from Euclidean spacial domains $U\subseteq \mathbb{R}^n$ which serve as a theory-encoding entity to $X$ and realizing the theory in a model. It is a sort of representation theory, the group being at least the diffeomorphism group of $X$, and $X$ being the ‘representation’. So syntax-semantics duality is in fact the same thing as the duality underlying dualities such as Tannaka-Krein.