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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2011
    • (edited Jan 25th 2011)

    Let \mathcal{E} be a local topos or sheaf topos over a concrete site and XX \in \mathcal{E} a concrete sheaf.

    The concreteness-condition on a sheaf may be read as saying that XX has “enough points”, in a sense.

    What about the slice topos /X\mathcal{E}/X? What can we say about its topos points? Under which conditions does it have enough?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJan 26th 2011

    Any slice topos of a topos with enough points has enough points. Proof: suppose EE has enough points. A point of E/XE/X is a point e:SetEe\colon Set \to E together with an element xe *(X)x\in e^*(X). To prove that it has enough points, we must show that if A,BE/XA,B\in E/X and f:ABf:A\to B is a morphism in E/XE/X such that (e,x) *(f)(e,x)^*(f) is an isomorphism for every point (e,x)(e,x) of E/XE/X , then ff is an isomorphism. But (e,x) *(e,x)^* is given by applying e *e^* and then taking the fiber over xx, so if (e,x) *(f)(e,x)^*(f) is an isomorphism for all ee and xx, then e *(f)e^*(f) is an isomorphism for all ee, hence (since EE has enough points) ff is an isomorphism in EE, hence also in E/XE/X .

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2011
    • (edited Jan 26th 2011)

    Thanks, Mike!

    I have added some discussion to over-topos. I see the points of the form (e,x)(e,x), they are the composites

    (e,x):Setx *Set/e *(X)e/X/X. (e,x) : Set \stackrel{x^*}{\to} Set/e^*(X) \stackrel{e/X}{\to} \mathcal{E}/X \,.

    How do I see that every point of the slice topos arises this way?

    But more importantly: in which generality can we assume \mathcal{E} to have enough points in the first place? To talk about concrete sheaves we need a local topos. What can we say about local toposes having enough points? (Certainly they have one important canonical point, by definition.)

    I need to think again about the standard example that I am interested in, the topos Sh(CartSp)Sh(Mfd)Sh(CartSp) \simeq Sh(Mfd). It should have enough points, with one point (n)(n) per nn \in \mathbb{N} given by the stalk at any ordinary point in n\mathbb{R}^n.

    Together with the above this would also show where my intuition about concrete sheaves inducing “enough points” was wrong: the topos points of Sh(CartSp)/XSh(CartSp)/X are not related to global points *X* \to X but to stalks of maps of disks into XX, hence manifestly do not distinguish between concrete and non-concrete XX.

    Is there maybe some other property of the slice /X\mathcal{E}/X that witnesses the fact that XX is concrete?

    Let’s see, if we start with a local topos (p *p *p !):Set(p^* \dashv p_* \dashv p^!) : \mathcal{E} \to Set then we should maybe be looking at the canonical point e 0:=(p *p !):Sete_0 := (p_* \dashv p^!) : Set \to \mathcal{E} and the induced points of the special form

    (e 0,x):Set/X. (e_0, x) : Set \to \mathcal{E}/X \,.

    Now these are really those given by the global points x:*Xx : * \to X of XX. So concreteness of XX should say that /X\mathcal{E}/X has in some sense enough of these e 0e_0-points.


    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2011
    • (edited Jan 26th 2011)

    Not sure if it helps, but just an observation: if \mathcal{E} is a local topos and XX \in \mathcal{E} a concrete object, then the “global points”-monad corresponding to the adjunction Set/p *(X)p !/Xp */X/XSet/p_*(X) \stackrel{\overset{p_*/X}{\leftarrow}}{\underset{p^!/X}{\to}} \mathcal{E}/X acts on an object (AX)/X(A \to X) \in \mathcal{E}/X by concretifying AA relative to XX:

    by construction of the slice geometric morphism (p */Xp !/X)(p_*/X \dashv p^!/X) we have that p !/Xp */X(AX) p^!/X \circ p_*/X (A \to X) is the pullback A˜\tilde A in

    A˜ p !p *A X p !p *X, \array{ \tilde A &\to& p^! p_* A \\ \downarrow && \downarrow \\ X &\to& p^! p_* X } \,,

    where the bottom morphism is the unit. By definition of concrete objects the bottom morphism is a mono precisely if XX is concrete. Since monos are stable under pullback, also the top morphism is a mono. Applying p *p_* to the top morphism and using that p !p^! is by definition full and faithful, we find that ΓA˜ΓA\Gamma \tilde A \hookrightarrow \Gamma A is a mono. By the universal property of the unit this implies that A˜p !p *A˜\tilde A \to p_! p^* \tilde A is a mono, hence that A˜\tilde A itself is concrete.

    Now, on the objects A˜X\tilde A \to X in the image of this “relative concretization”-monad it is certainly true that the (e 0,x)(e_0,x)-points detect isomorphisms.


    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJan 26th 2011

    How do I see that every point of the slice topos arises this way?

    By pullback, it suffices to show that every section of E/XEE/X\to E arises from a map 1X1\to X in E. You can regard that as a special case of the full embedding of internal locales in E into localic geometric morphisms over E, where X is regarded as a discrete internal locale in E.