nLab smooth structure on a topos

Idea

A topos can be viewed as a generalization of a topological space. A smooth structure on a topos is a corresponding generalization of the notion of smooth manifold, in that to put a smooth structure on $Sh(X)$, for a topological space $X$, is (at least closely related to) putting the structure of a smooth manifold on $X$. However, since it is phrased internally with reference to the (Cauchy and Dedekind) real numbers objects, it is applicable to any topos.

Definition

Given a topos $\mathbf{H}$, write

• $\mathbb{N} \in \mathbf{H}$ for its natural numbers object;

• $\mathbb{R}_{C} \in \mathbf{H}$ for its Cauchy real numbers object;

• $\mathbb{R}_{D} \in \mathbf{H}$ for its Dedekind real numbers object;

• $\mathbb{R} \in$ Set for the external real numbers.

There are canononical subobject inclusions

$$\mathbb{N} \hookrightarrow \mathbb{R}_C \hookrightarrow \mathbb{R}_D \,.$$

Definition

(Fourman 75, def. 3.6) A morphism

$$g \colon \mathbb{R}_D^{n_1} \longrightarrow \mathbb{R}_D^{n_2}$$

in the topos $\mathbf{H}$ is called a standard function, precisely if

1. it is a continuous function in the internal sense that

   $$\underset{\epsilon \gt 0}{\forall} \underset{\delta \gt 0}{\exists} \underset{\vec x, \vec y \in \mathbb{R}_D^{n_1}}{\forall} \left( \left( \underset{i}{max} {\vert x_i - y_i \vert} \lt \delta \right) \Rightarrow \left( \underset{i}{max} {\vert g(\vec x)_i - g(\vec y)_i \vert} \lt \epsilon \right) \right)$$

2. it respects the sub-object of Cauchy reals, in that

   $$\underset{\vec x \in \mathbb{R}_C^{n_1}}{\forall} \left( g(\vec x) \in \mathbb{R}_{C}^{n_2} \right) \,.$$

We will furthermore consider smooth standard functions, meaning standard functions that satisfy the internalized ordinary definition of smooth function (i.e. $C^\infty$).

We define an equivalence relation on $\mathbb{R}_D^n$ by taking two elements to be equivalent if there are smooth standard functions taking them into each other:

$$\vdash (r \simeq s) \coloneqq \underset{{f,g } \atop {smooth \; standard\;functions}}{\exists} \left( \left( f(r) = s \right) \wedge \left( g(s) = r \right) \right) \,.$$

Definition

(Fourman 75, def. 4.1) A smooth structure of dimension $n$ on a topos $\mathbf{H}$ is an equivalence class for the above equivalence relation on $\mathbb{R}_D^n$. In other words, the object of smooth structures of dimension $n$ is the quotient of $\mathbb{R}_D^n$ by this equivalence relation.

Definition

(Fourman 75, def. 4.4) Given a smooth structure $S$ of dimension $n$ on the topos $\mathbf{H}$ according to def. , then the smooth real number object is the subobject

$$\mathbb{R}_S \hookrightarrow \mathbb{R}_D$$

defined internally as the set of all images of points in $S\subseteq \mathbb{R}_D^n$ under smooth standard functions $\mathbb{R}_D^n \to \mathbb{R}_D$. In symbols, it is given by

$$\mathbb{R}_S \coloneqq \left\{ x \in \mathbb{R}_D \;|\; \underset{{smooth \; standard\;funct.}\atop{\mathbb{R}_D^n \stackrel{f}{\to} \mathbb{R}_D}}{\exists} \underset{s \in S}{\exists} (x = f(s)) \right\} \,.$$

Note that any function constant at a Cauchy real is standard. Therefore, every Cauchy real is a smooth real.

Examples

Example

(Fourman 75, example 4.3 1) Let $X$ be a smooth manifold of dimension $n$, with sheaf topos $\mathbf{H} \coloneqq Sh(X)$. As shown at real numbers object, $\mathbb{R}_D$ is then the sheaf of continuous real-valued functions on $X$.

Let $S\subseteq \mathbb{R}_D^n$ be the sheaf of local coordinate systems, i.e. $S(U)$ is the set of real-valued functions $U\to \mathbb{R}^n$ that are smooth and are locally diffeomorphisms onto their images. Then $S$ is a smooth structure on $Sh(X)$ of dimension $n$, according to def. .

The corresponding object $\mathbb{R}_S$ of smooth reals is the sheaf of smooth real-valued functions $X\to \mathbb{R}$. Note that since $X$ is locally connected, the Cauchy real numbers object $\mathbb{R}_C$ in $Sh(X)$ is the sheaf of locally constant real-valued functions, so $\mathbb{R}_S$ sits strictly in between $\mathbb{R}_C$ and $\mathbb{R}_D$.

Example

Let $h:1\to \mathbb{R}_D^n$ be any global section of $\mathbb{R}_D^n$. Then the equivalence class of $h$, under the above equivalence relation, is a smooth structure. In particular, if $h$ is a tuple of Cauchy real numbers (such as $0$), then so is every point in $S$, and thus $\mathbb{R}_S = \mathbb{R}_C$. This gives the “discrete” smooth structure on $\mathbf{H}$.

For $\mathbf{H}=Sh(X)$, such a global section is a continuous map $h:X\to \mathbb{R}^n$, and this gives the smooth structure “cogenerated by” $h$, in that it makes $h$ smooth and only those other functions that must be smooth if $h$ is. The case when $h$ is a Cauchy real corresponds to $h:X\to \mathbb{R}^n$ being locally constant, in which case all smooth functions are locally constant; this is the “minimal” smooth structure on a topological space $X$.

References

• Michael Fourman, Comparaison des Réels d’un Topos - Structures Lisses sur un Topos Elémentaire , Cah. Top. Géom. Diff. Cat. 16 (1975) pp.233-239. ( Colloque Amiens 1975 proceedings ) (p. 18-24 in NUMDAM))