smooth structure on a topos



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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)Sh(X), for a topological space XX, is (at least closely related to) putting the structure of a smooth manifold on XX. However, since it is phrased internally with reference to the (Cauchy and Dedekind) real numbers objects, it is applicable to any topos.


Given a topos H\mathbf{H}, write

There are canononical subobject inclusions

C D. \mathbb{N} \hookrightarrow \mathbb{R}_C \hookrightarrow \mathbb{R}_D \,.

(Fourman 75, def. 3.6) A morphism

g: D n 1 D n 2 g \colon \mathbb{R}_D^{n_1} \longrightarrow \mathbb{R}_D^{n_2}

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

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

    ϵ>0δ>0x,y D n 1((maxi|x iy i|<δ)(maxi|g(x) ig(y) i|<δ)) \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 \delta \right) \right)
  2. it respects the sub-object of Cauchy reals, in that

    x C n 1(g(x) C n 2). \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 C^\infty).

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

(rs)f,gsmoothstandardfunctions((f(r)=s)(g(s)=r)). \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) \,.

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


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

S D \mathbb{R}_S \hookrightarrow \mathbb{R}_D

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

S{x D|smoothstandardfunct. D nf DsS(x=f(s))}. \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.



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

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

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


Let h:1 D nh:1\to \mathbb{R}_D^n be any global section of D n\mathbb{R}_D^n. Then the equivalence class of hh, under the above equivalence relation, is a smooth structure. In particular, if hh is a tuple of Cauchy real numbers (such as 00), then so is every point in SS, and thus S= C\mathbb{R}_S = \mathbb{R}_C. This gives the “discrete” smooth structure on H\mathbf{H}.

For H=Sh(X)\mathbf{H}=Sh(X), such a global section is a continuous map h:X nh:X\to \mathbb{R}^n, and this gives the smooth structure “cogenerated by” hh, in that it makes hh smooth and only those other functions that must be smooth if hh is. The case when hh is a Cauchy real corresponds to h:X nh: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 XX.


  • 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))

Last revised on November 10, 2014 at 12:09:50. See the history of this page for a list of all contributions to it.