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Let $\{\mathbb{R}^1\} \hookrightarrow S \hookrightarrow Top$ be a small full subcategory of that of topological spaces, such that it includes the real line, and let $\mathrm{Sh}(S)$ the sheaf topos over $S$.
What are sufficient conditions on $S$ which guarantee that the internal real line of $\mathrm{Sh}(S)$ is represented by the external real line?
Specifically, is it the case for $S$ being
a small version of locally contractible topological spaces?
the category of topological manifolds?
Mac Lane and Moerdijk discuss this in the section about Brouwer’s theorem on continuous functions. They consider small full subcategories $\mathbf{T} \subseteq \mathbf{Top}$ satisfying these conditions:
They then prove that the Dedekind real numbers in $\mathbf{Sh}(\mathbf{T})$ is the sheaf represented by $\mathbb{R}$.
So are locally contractible spaces closed under finite limits? Topological manifolds certainly aren’t.
First guess at a counterexample: given the constant function $0\colon \mathbb{R}^2 \to \mathbb{R}$ and the function that measures the distance to the Hawaiian earring, $d(-,H)\colon \mathbb{R}^2 \to \mathbb{R}$. Then $H \to \mathbb{R}^2 \rightrightarrows \mathbb{R}^2$ is a equaliser, is it not? $H$ is of course famously not locally contractible.
Scanning M&ML’s proof, though, there is nothing explicitly using the finite completeness. I suspect finite products and closure under pullback of covers may be sufficient. They use the open cover topology, but perhaps by ’covers’ one could take some other class of maps/covering families (e.g. open surjections) under which to close up the small category under pullbacks, rather than all maps.
That sounds promising.
Thanks everyone!
Now I had time to look into this.
So the proof is on p. 328. It works essentially by arguing that over any object in the site, the argument reduces to that for the real numbers in the petit sheaf topos on that object. That is the case proven earlier on pages 323-325.
Looking at this, my impression is, as David says, that finite limits in the site is never invoked except that it guarantees the induced Grothendieck topology. But if we just used an induced coverage structure instead, the argument would seem to go through essentially unchanged.
It seems therefore that the answer to my two questions is Yes and Yes.
But let me know if I am missing something.
Here is a slicker and more general argument, based on D4.7.6 in the Elephant.
The sheaf topos $Sh(\mathbb{R})$ is the classifying topos of the geometric theory of a real number, in the sense that geometric morphisms $E \to Sh(\mathbb{R})$ are equivalent to global points of the real numbers object $\mathbb{R}_E$ in $E$. Since pullback functors are logical, they preserve the real numbers object; thus for any $X\in E$, maps $X\to \mathbb{R}_E$ are equivalent to geometric morphisms $E/X \to Sh(\mathbb{R})$. But $Sh(\mathbb{R})$ is localic, so such geometric morphisms factor through the localic reflection of $E/X$, and therefore are equivalent to continuous $\mathbb{R}$-valued functions defined on the “little locale of $X$”, i.e. the locale associated to the frame of subobjects of $X$ in $E$.
Therefore, if $E = Sh(S)$ for some site $S$, then $\mathbb{R}_E$ is the sheaf on $S$ where $\mathbb{R}_E(X)=$ the set of continuous $\mathbb{R}$-valued functions on the little locale of $y X \in E$. So it suffices to observe that if $S\subset Top$ is closed under open subspaces and equipped with the open-cover coverage, then every subobject of $y X\in Sh(S)$, for any $X\in S$, is uniquely representable by an open subset of $X$.
I’ve recorded this argument at real numbers object. It has the additional advantage of suggesting a way to characterize the real number object in any sheaf topos. For instance, what is the little locale of a smooth locus?
Thanks! Excellent.
Wait, now I don’t believe this any more:
every subobject of $y X\in Sh(S)$, for any $X\in S$, is uniquely representable by an open subset of $X$.
What about, say, the subsheaf consisting of all maps $Y\to X$ in $S$ that are constant at some $x\in X$?
Now I don’t see why to believe the theorem any more either. In terms of the proof in ML&M VI.9.2, this corresponds to asking why the composite $(L,U) \mapsto f_{L,U} \mapsto (L_{f_{L,U}},U_{f_{L,U}})$ is the identity (which they give no argument for), since the definition of $f_{L,U}$ refers only to the inclusions of open subspaces of $W$ and ignores the values of $L$ and $U$ on all other maps into $W$. Anyone help?
Wait, now I don’t believe this any more:
every subobject of $y X\in Sh(S)$, for any $X\in S$, is uniquely representable by an open subset of $X$.
An easier counterexample is obtained by noting that any morphism $1 \to y X$ is a monomorphism, but points are surely not open subspaces of arbitrary $X$. I previously asked a similar question about the localic reflection of the Zariski topos.
However, perhaps the canonical map $\mathbf{Loc}(X, \mathbb{R}) \to \mathbf{Loc}(Sub(y X), \mathbb{R})$ is a bijection for nice $X$?
Isn’t that the same as my example, where $1\to y X$ picks out the $x\in X$ in question?
Here’s a little observation. When $\mathbf{T}$ is nice enough (e.g. satisfying the hypotheses of Mac Lane and Moerdijk, but less suffices), $\mathbf{Sh}(\mathbf{T}_{/ X})$ is local over $\mathbf{Sh} (X)$, so every global real number in $\mathbf{Sh} (X)$ is obtained as the restriction of some global real number in $\mathbf{Sh}(\mathbf{T}_{/ X})$. I believe that this is actually a bijection: because if $L$ is a topological space with a focal point, then every continuous map $L \to \mathbb{R}$ is constant.
Concerning the role of local maps for the reals, I would like to point to C3.6.11 (p.703). I was wondering whether this property holds as well for the ’colocal’=totally connected maps ? my guess considering p.709 there is yes. Johnstone points also out there that the classifier for reals is a very special sort of locale.
I would like to say also when locality is all that is needed there, it seems that the remarks p.577-8 where Johnstone discusses the gros site of spaces, would yield the same result for manifolds and schemes because Johnstone says that the examples e,f) on p.77 would permit a similar construction with appropriate local maps.
Excellent! That fills the gap perfectly.
Brilliant, thanks Thomas! Would you like to answer the MO question?
To be honest, I only had a vague idea after reading Zhen Lin’s comment that this fits your bill. I just happen to have come across this in order to figure out how far this apllies to totally connected morphisms. A lot of the concepts involved I find hard to swallow like the eg. ’grouplike topos’ .
The ’homotopy equivalence’ between gros sh(T)/yX and sh(X) which presumably is implicit in Johnstone’s result as well and appears to fly in the face of ’qualtitative distinctions’ is also something which profoundly troubles me ever since I’ve come across the passage in MM (p.416).
So I guess, to clarify your MO-question demands better grasp of the concepts involved than I can offer.
It actually reduces to something quite simple in this case, as I explain.
The idea that local geometric morphisms are homotopy equivalences is a special case of the idea that adjoint pairs of geometric morphisms are homotopy equivalences, which really just comes down to the observation that Sierpiński space $\Sigma$ admits a endpoint-preserving map $[0, 1] \to \Sigma$. But the “shape” of a topos as captured by homotopy is a rather coarse invariant.
Excellent. I have added pointer to this at real number object. You should go and add your lemma there.
@Zhen Lin: well my worries about this homotopy equivalence actually started since I tried to let the Lawvere program grow on me. So probably, my thoughts were/are a bit too rule-of-thumby: when your cohomology fails to detect a difference between gros/petit than there is something wrong with either the gros-petit distinction or the cohomology theory.
Thomas, I would say that cohomology is designed to be a very coarse invariant. In classical algebraic topology, it notices only the homotopy type of a space, discarding all information about homeomorphism type; but that doesn’t mean it’s “wrong”. (-:
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