Assuming some extra structure we simplify the characterization of the categories with finite limits whose exact completions are toposes given in Menni . This simplification allows us to obtain new examples and non-examples and also to provide a new perspective and an alternative proof of recent results on the inevitability of untypedness for realizability toposes.

We give the natural topological model for $\neg$BD-${\mathbb N}$, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-${\mathbb N}$. Also, the natural topological model for $\neg$BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-$\mathbb{N}$, it is brought out in detail how BD-$\mathbb N$ fails.

In [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$. We then show that adding (...) such choice principles does not change the arithmetic part of either $\mathsf {CZF}$ or $\mathsf {IZF}$. (shrink)