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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 25th 2011
   • (edited Jan 25th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexLet $\mathcal{E}$ be a [[local topos]] or sheaf topos over a [[concrete site]] and $X \in \mathcal{E}$ a [[concrete sheaf]]. The concreteness-condition on a sheaf may be read as saying that $X$ has "enough points", in a sense. What about the slice topos $\mathcal{E}/X$? What can we say about its [[point of a topos|topos points]]? Under which conditions does it have enough?

   Let be a local topos or sheaf topos over a concrete site and a concrete sheaf.

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

   What about the slice topos ? What can we say about its topos points? Under which conditions does it have enough?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 26th 2011
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   Author: Mike Shulman
   Format: MarkdownItexAny slice topos of a topos with enough points has enough points. Proof: suppose $E$ has enough points. A point of $E/X $ is a point $e\colon Set \to E$ together with an element $x\in e^*(X)$. To prove that it has enough points, we must show that if $A,B\in E/X$ and $f:A\to B$ is a morphism in $E/X $ such that $(e,x)^*(f)$ is an isomorphism for every point $(e,x)$ of $E/X $, then $f$ is an isomorphism. But $(e,x)^*$ is given by applying $e^*$ and then taking the fiber over $x$, so if $(e,x)^*(f)$ is an isomorphism for all $e$ and $x$, then $e^*(f)$ is an isomorphism for all $e$, hence (since $E$ has enough points) $f$ is an isomorphism in $E$, hence also in $E/X $.

   Any slice topos of a topos with enough points has enough points. Proof: suppose has enough points. A point of is a point together with an element . To prove that it has enough points, we must show that if and is a morphism in such that is an isomorphism for every point of , then is an isomorphism. But is given by applying and then taking the fiber over , so if is an isomorphism for all and , then is an isomorphism for all , hence (since has enough points) is an isomorphism in , hence also in .

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 26th 2011
   • (edited Jan 26th 2011)
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   Author: Urs
   Format: MarkdownItexThanks, Mike! I have added some discussion to [[over-topos]]. I see the points of the form $(e,x)$, they are the composites $$ (e,x) : Set \stackrel{x^*}{\to} Set/e^*(X) \stackrel{e/X}{\to} \mathcal{E}/X \,. $$ 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 $Sh(CartSp) \simeq Sh(Mfd)$. It should have enough points, with one point $(n)$ per $n \in \mathbb{N}$ given by the stalk at any ordinary point in $\mathbb{R}^n$. Together with the above this would also show where my intuition about concrete sheaves inducing "enough points" was wrong: the topos points of $Sh(CartSp)/X$ are not related to global points $* \to X$ but to stalks of maps of disks into $X$, hence manifestly do not distinguish between concrete and non-concrete $X$. Is there maybe some other property of the slice $\mathcal{E}/X$ that witnesses the fact that $X$ is concrete? Let's see, if we start with a local topos $(p^* \dashv p_* \dashv p^!) : \mathcal{E} \to Set$ then we should maybe be looking at the canonical point $e_0 := (p_* \dashv p^!) : Set \to \mathcal{E}$ and the induced points of the special form $$ (e_0, x) : Set \to \mathcal{E}/X \,. $$ Now _these_ are really those given by the global points $x : * \to X$ of $X$. So concreteness of $X$ should say that $\mathcal{E}/X$ has in some sense enough of these $e_0$-points. Hm...

   Thanks, Mike!

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

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

   But more importantly: in which generality can we assume 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 . It should have enough points, with one point per given by the stalk at any ordinary point in .

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

   Is there maybe some other property of the slice that witnesses the fact that is concrete?

   Let’s see, if we start with a local topos then we should maybe be looking at the canonical point and the induced points of the special form

   Now these are really those given by the global points of . So concreteness of should say that has in some sense enough of these -points.

   Hm…

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 26th 2011
   • (edited Jan 26th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexNot sure if it helps, but just an observation: if $\mathcal{E}$ is a local topos and $X \in \mathcal{E}$ a concrete object, then the "global points"-monad corresponding to the adjunction $Set/p_*(X) \stackrel{\overset{p_*/X}{\leftarrow}}{\underset{p^!/X}{\to}} \mathcal{E}/X$ acts on an object $(A \to X) \in \mathcal{E}/X$ by concretifying $A$ relative to $X$: by construction of the slice geometric morphism $(p_*/X \dashv p^!/X)$ we have that $ p^!/X \circ p_*/X (A \to X)$ is the pullback $\tilde A$ in $$ \array{ \tilde A &\to& p^! p_* A \\ \downarrow && \downarrow \\ X &\to& p^! p_* X } \,, $$ where the bottom morphism is the unit. By definition of concrete objects the bottom morphism is a mono precisely if $X$ is concrete. Since monos are stable under pullback, also the top morphism is a mono. Applying $p_*$ to the top morphism and using that $p^!$ is by definition full and faithful, we find that $\Gamma \tilde A \hookrightarrow \Gamma A$ is a mono. By the universal property of the unit this implies that $\tilde A \to p_! p^* \tilde A $ is a mono, hence that $\tilde A$ itself is concrete. Now, on the objects $\tilde A \to X$ in the image of this "relative concretization"-monad it is certainly true that the $(e_0,x)$-points detect isomorphisms. Hm...

   Not sure if it helps, but just an observation: if is a local topos and a concrete object, then the “global points”-monad corresponding to the adjunction acts on an object by concretifying relative to :

   by construction of the slice geometric morphism we have that is the pullback in

   where the bottom morphism is the unit. By definition of concrete objects the bottom morphism is a mono precisely if is concrete. Since monos are stable under pullback, also the top morphism is a mono. Applying to the top morphism and using that is by definition full and faithful, we find that is a mono. By the universal property of the unit this implies that is a mono, hence that itself is concrete.

   Now, on the objects in the image of this “relative concretization”-monad it is certainly true that the -points detect isomorphisms.

   Hm…

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJan 26th 2011
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   Author: Mike Shulman
   Format: MarkdownItex> How do I see that every point of the slice topos arises this way? By pullback, it suffices to show that every section of $E/X\to E$ arises from a map $1\to X$ 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.

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

   By pullback, it suffices to show that every section of arises from a map 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.