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A topos is a category that behaves very much like the category of sets and possesses a good notion of localization. Related to topos are: sheaves, presheaves, descent, stacks, localization,...

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

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 …