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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.)

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    $\begingroup$ I would say that there is several notion that can be introduced that answer the question, but no real criterion to say that one is better than others. You can ask for the localic reflection to be regular. Or ask that every object of the topos can be covered by an object $X$ such that $\mathcal{T}_{/X}$ has a regular localic reflection. Or ask stronger conditions that will also implies that the topos is separated (proper diagonal). $\endgroup$ – Simon Henry Dec 30 '16 at 21:41
  • $\begingroup$ Yes, I thought of those, but none of them is really topos-theoretic. One good criterion for correctness would be that it can be categorified, the way proper categorifies to tidy and open categorifies to locally connected. $\endgroup$ – Mike Shulman Dec 31 '16 at 21:23
  • $\begingroup$ A remark that might be useful but that I haven't been able to use: the relation $U \triangleleft V$ that appears in the definition of regularity is equivalent to "internally, there exists a finite object $F$ such that $U \subset F \subset V$ ". Moreover, finite objects in toposes are closely related to the Hausdorff conditions. $\endgroup$ – Simon Henry Jan 3 '17 at 13:49
  • $\begingroup$ @SimonHenry That's interesting! It's not obvious to me why that should be; can you explain (maybe in chat)? $\endgroup$ – Mike Shulman Jan 3 '17 at 20:52
  • $\begingroup$ For subterminal objects $U $ and $V$ having internally "There exists $F $ kuratowski finite such that $ U \subset F \subset V $ " is equivalent to " U is empty or V is inhabited " depending on whether the $n$ appearing in the definition of finiteness is zero or non zero. But "U is empty or V is inhabited " is "$\neg U \cup V$" saying that it is true is a possible definition of $U \triangleleft V $. $\endgroup$ – Simon Henry Jan 3 '17 at 23:20
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As far as I am aware, there is not. (Unless, by definition, you want to say that a topos is regular if its localic reflection is regular.) Not sure if that's helpful!

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