What does an ideal correspond to in the internal language of sheaves?

Question

Suppose I have a sheaf $\mathcal F$ in some topos $\mathrm{Sh}(\mathcal C)$. Then this becomes the sheaf of rings from algebraic geometry when described as a ring in the internal language of the topos. Whilst in the internal language we were to describe a ring ideal, what would this correspond to externally?

Answer 1

I assume that you mean that $\mathcal{F}$ is a sheaf of rings.

What's internally an ideal of $\mathcal{F}$ is externally simply a sheaf of ideals.

In case that the topos in question is the little Zariski topos of a scheme and $\mathcal{F}$ is the structure sheaf $\mathcal{O}_X$, you could ask a follow-up question: How can we characterize quasicoherent sheaves of ideals in the internal language?

The answer is that a sheaf $\mathcal{I}$ of ideals is quasicoherent if and only if, from the internal point of view, $$ \forall f : \mathcal{O}_X. \forall s : \mathcal{O}_X. (\text{$f$ invertible} \Rightarrow s \in \mathcal{I}) \Longrightarrow \bigvee_{n \geq 0} (f^n s \in \mathcal{I}). $$ A proof of this characterization is given in Section 8 of these notes of mine, more precisely Corollary 8.5.