A topological space $X$ is regular if a point not belonging to a closed set can be separated from it by disjoint opens. Equivalently, this means if $x\in U$ for an open set $U$, then there are opens $V$ and $W$ with $x\in W$, $V\cap W = \emptyset$, and $U \cup W = X$.

More generally, a locale is regular if every open $U$ is the join of opens $V$ that are "well-inside" it, where $V$ is well-inside $U$ if there exists an open $W$ such that $V\cap W = \emptyset$ and $U \cup W = X$.

Often, topological properties of locales can be generalized naturally to toposes (or, often, to geometric morphisms). For instance, a locale is Hausdorff if its diagonal $X\to X\times X$ is a proper map, and this generalizes to toposes since we know what a proper map of toposes is.

Is there a notion of "regularity" for toposes that generalizes the topological notion of regularity for spaces and locales?

(I'm being careful with terminology because it would presumably have nothing to do with the extant notion of "regular topos", which means a topos of sheaves for the regular-epimorphism topology on a regular category.)

1more comment