## General Idea

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

*iff*the following three equivalent conditions are satisfied

- (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.
- (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.
- (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.

This says that

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

*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:

- (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. - (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 $\mathcal{F}_Y$. - (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.