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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2014
    • (edited Oct 26th 2014)

    Let { 1}STop\{\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 Sh(S)\mathrm{Sh}(S) the sheaf topos over SS.

    What are sufficient conditions on SS which guarantee that the internal real line of Sh(S)\mathrm{Sh}(S) is represented by the external real line?

    Specifically, is it the case for SS being

    1. a small version of locally contractible topological spaces?

    2. the category of topological manifolds?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeOct 26th 2014

    Mac Lane and Moerdijk discuss this in the section about Brouwer’s theorem on continuous functions. They consider small full subcategories TTop\mathbf{T} \subseteq \mathbf{Top} satisfying these conditions:

    • T\mathbf{T} is closed under finite limits.
    • T\mathbf{T} is closed under open subspaces.
    • T\mathbf{T} contains \mathbb{R}.

    They then prove that the Dedekind real numbers in Sh(T)\mathbf{Sh}(\mathbf{T}) is the sheaf represented by \mathbb{R}.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeOct 27th 2014

    So are locally contractible spaces closed under finite limits? Topological manifolds certainly aren’t.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 27th 2014

    First guess at a counterexample: given the constant function 0: 20\colon \mathbb{R}^2 \to \mathbb{R} and the function that measures the distance to the Hawaiian earring, d(,H): 2d(-,H)\colon \mathbb{R}^2 \to \mathbb{R}. Then H 2 2H \to \mathbb{R}^2 \rightrightarrows \mathbb{R}^2 is a equaliser, is it not? HH is of course famously not locally contractible.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 27th 2014

    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.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeOct 27th 2014

    That sounds promising.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2014
    • (edited Oct 27th 2014)

    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.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeOct 31st 2014
    • (edited Oct 31st 2014)

    Here is a slicker and more general argument, based on D4.7.6 in the Elephant.

    The sheaf topos Sh()Sh(\mathbb{R}) is the classifying topos of the geometric theory of a real number, in the sense that geometric morphisms ESh()E \to Sh(\mathbb{R}) are equivalent to global points of the real numbers object E\mathbb{R}_E in EE. Since pullback functors are logical, they preserve the real numbers object; thus for any XEX\in E, maps X EX\to \mathbb{R}_E are equivalent to geometric morphisms E/XSh()E/X \to Sh(\mathbb{R}). But Sh()Sh(\mathbb{R}) is localic, so such geometric morphisms factor through the localic reflection of E/XE/X, and therefore are equivalent to continuous \mathbb{R}-valued functions defined on the “little locale of XX”, i.e. the locale associated to the frame of subobjects of XX in EE.

    Therefore, if E=Sh(S)E = Sh(S) for some site SS, then E\mathbb{R}_E is the sheaf on SS where E(X)=\mathbb{R}_E(X)= the set of continuous \mathbb{R}-valued functions on the little locale of yXEy X \in E. So it suffices to observe that if STopS\subset Top is closed under open subspaces and equipped with the open-cover coverage, then every subobject of yXSh(S)y X\in Sh(S), for any XSX\in S, is uniquely representable by an open subset of XX.

    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?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2014

    Thanks! Excellent.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeOct 31st 2014

    Wait, now I don’t believe this any more:

    every subobject of yXSh(S)y X\in Sh(S), for any XSX\in S, is uniquely representable by an open subset of XX.

    What about, say, the subsheaf consisting of all maps YXY\to X in SS that are constant at some xXx\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)f L,U(L f L,U,U f L,U)(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,Uf_{L,U} refers only to the inclusions of open subspaces of WW and ignores the values of LL and UU on all other maps into WW. Anyone help?

    • CommentRowNumber11.
    • CommentAuthorZhen Lin
    • CommentTimeOct 31st 2014
    • (edited Oct 31st 2014)

    Wait, now I don’t believe this any more:

    every subobject of yXSh(S)y X\in Sh(S), for any XSX\in S, is uniquely representable by an open subset of XX.

    An easier counterexample is obtained by noting that any morphism 1yX1 \to y X is a monomorphism, but points are surely not open subspaces of arbitrary XX. I previously asked a similar question about the localic reflection of the Zariski topos.

    However, perhaps the canonical map Loc(X,)Loc(Sub(yX),)\mathbf{Loc}(X, \mathbb{R}) \to \mathbf{Loc}(Sub(y X), \mathbb{R}) is a bijection for nice XX?

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeOct 31st 2014

    Isn’t that the same as my example, where 1yX1\to y X picks out the xXx\in X in question?

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeNov 1st 2014
    • (edited Nov 1st 2014)
    • CommentRowNumber14.
    • CommentAuthorZhen Lin
    • CommentTimeNov 3rd 2014

    Here’s a little observation. When T\mathbf{T} is nice enough (e.g. satisfying the hypotheses of Mac Lane and Moerdijk, but less suffices), Sh(T /X)\mathbf{Sh}(\mathbf{T}_{/ X}) is local over Sh(X)\mathbf{Sh} (X), so every global real number in Sh(X)\mathbf{Sh} (X) is obtained as the restriction of some global real number in Sh(T /X)\mathbf{Sh}(\mathbf{T}_{/ X}). I believe that this is actually a bijection: because if LL is a topological space with a focal point, then every continuous map LL \to \mathbb{R} is constant.

    • CommentRowNumber15.
    • CommentAuthorThomas Holder
    • CommentTimeNov 3rd 2014
    • (edited Nov 3rd 2014)

    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.

    • CommentRowNumber16.
    • CommentAuthorZhen Lin
    • CommentTimeNov 3rd 2014

    Excellent! That fills the gap perfectly.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeNov 3rd 2014

    Brilliant, thanks Thomas! Would you like to answer the MO question?

    • CommentRowNumber18.
    • CommentAuthorThomas Holder
    • CommentTimeNov 4th 2014

    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.

    • CommentRowNumber19.
    • CommentAuthorZhen Lin
    • CommentTimeNov 4th 2014

    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 Σ\Sigma admits a endpoint-preserving map [0,1]Σ[0, 1] \to \Sigma. But the “shape” of a topos as captured by homotopy is a rather coarse invariant.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2014

    Excellent. I have added pointer to this at real number object. You should go and add your lemma there.

    • CommentRowNumber21.
    • CommentAuthorThomas Holder
    • CommentTimeNov 4th 2014

    @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.

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeNov 5th 2014

    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”. (-: