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## Do functions defined on global elements give rise to arrows in a well-pointed topos?

Hi all,

Sorry if this question is not the right level for mathoverflow, but I already tried math.stackexchange and received no answers.

Suppose that $\mathcal{E}$ is a well-pointed elementary topos, that $X$ and $Y$ are objects of $\mathcal{E}$, and that $F$ is a function which maps global elements $p: 1 \to X$ to global elements $F(p): 1 \to Y$ (here $1$ is the terminal object of $\mathcal{E}$). Does there exist a (necessarily unique) arrow $f: X \to Y$ in $\mathcal{E}$ such that $fp = F(p)$ for all $p$? Equivalently, is any object in a well-pointed topos the coproduct over its global elements of $1$? It's easy to show that the answer is "yes" if the coproduct exists since the induced map $\coprod_{p \in \Gamma X} 1 \to X$ is iso. But I don't know whether the coproduct exists in general.

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 You are asking: given a map of sets, F:E(1,X) -> E(1,Y), is there an arrow X ->Y? If from F you can cook up a natural transformation E(-,X) -> E(-,Y), apply Yoneda and you are done. – David Roberts Oct 14 2010 at 22:31 I considered trying to use Yoneda, but I think it just takes me back where I started - constructing such a natural transformation $\sigma$ would in particular involve constructing the image $\sigma_X (1_X) \in \mathcal{R}(X, Y)$, which is what I need. – Phil Wild Oct 14 2010 at 22:43 Once you mention words like topos, I can hardly imagine that you are talking kinder-level of mathunderflow (math.stackexchange), but rather for here (unless one is throwing words without understanding). – Zoran Škoda Oct 16 2010 at 15:36 (sorry for the slow reply, only just noticed your comment) Well the question arose in the course of trying to understand a book that everyone working in the field has probably already read and understood, so I didn't think it would qualify as "research level". But thanks, it's good to know I'm not wasting MO's time. – Phil Wild Oct 20 2010 at 22:58

Here's a counterexample. Take $\mathcal E$ to be the topos of sets and functions of some countable model of ZFC (or a suitable weaker set theory, if you're worried that ZFC might be inconsistent). This is a well-pointed topos. The natural-number object $N$ in $\mathcal E$ has a countable infinity of global elements. So the number of functions (in the real world, not in the countable model) from global elements of $N$ to global elements of $N$ is the cardinal of the continuum. Only countably many of these correspond to morphisms in $\mathcal E$ from $N$ to $N$, because these morphisms are elements of your countable model.
In proving that the so-defined sets satisfy extensionality, the authors suppose that every point (i.e. node covered by the root) of a tree $T_1$ is isomorphic to some point of $T_2$, and then state that this gives a morphism from the subobject of points of $T_1$ to that of $T_2$. Do you know if the proof is wrong, or can the relevant morphisms be shown to exist in this case? If it's wrong, can it be fixed? – Phil Wild Oct 14 2010 at 23:01