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.



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


