Is there a Grothendieck topos $F$ such that, for any Grothendieck topos $E$, the category of geometric morphisms $$E\rightarrow F$$ is equivalent to the category of locales internal to $F$?

I suspect that such an $F$ doesn't exist, because the definition of locale doesn't look to me like it can be formalised as a geometric theory. But it would be nice if there was such a classifying space, since the locales internal to a topos are interesting objects. In particular, if $X$ is a locale then the locales internal to $\mathrm{Sh}(X)$ correspond bijectively with bundles over $X$, i.e. pairs $(Y,f)$ where $Y$ is a locale and $f:Y\to X$.

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    $\begingroup$ Excellent question. I too suspect that such an $F$ doesn't exist. If you're satisfied with essential surjectivity instead of equivalence, and if you're okay with switching to the dual category, then I can propose a substitute: the classifying topos of frame presentations. Every frame arises from a frame presentation, though not uniquely so. Details in frame presentations are for instance in a fine paper by Steve Vickers on the double powerlocale, Section 5, there called GRD system and cast in the language of geometric logic. $\endgroup$ – Ingo Blechschmidt Sep 28 '18 at 8:04
  • $\begingroup$ @IngoBlechschmidt The following occurred to me, I guess it must be well known. The internal locales do respect pullback: if $E\to F$ is a geometric morphism then we get a canonical functor from the category of locales internal to F to the category of locales internal to E. In other words we have a functor $\mathrm{Locale}:\mathbf{Topos}^\text{op}\to\mathbf{Cat}$. $\endgroup$ – Oscar Cunningham Oct 3 '18 at 11:09
  • $\begingroup$ @IngoBlechschmidt But $\mathbf{Topos}^\text{op}\to\mathbf{Cat}$ is the free cocompletion of $\mathbf{Topos}$, and $\mathbf{Topos}$ is already cocomplete. So there should be some sort of canonical "geometriztation" of $\mathrm{Locale}$ achieved by applying the functor $(\mathbf{Topos}^\text{op}\to\mathbf{Cat})\to\mathbf{Topos}$. Can we describe what this is? $\endgroup$ – Oscar Cunningham Oct 3 '18 at 11:09
  • $\begingroup$ Maybe there are size issues... $\endgroup$ – Oscar Cunningham Oct 4 '18 at 10:43
  • $\begingroup$ Ah, I see. The problem is that since $\mathbf{Topos}$ is large, it's free small-cocompletion isn't $\mathbf{Topos}^\text{op}\to\mathbf{Cat}$. $\endgroup$ – Oscar Cunningham Oct 5 '18 at 14:36

Points of a topos are always accessible, the categories of frames or locales are not.

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  • $\begingroup$ That sounds right, but could you give a sketch proof or reference? $\endgroup$ – Oscar Cunningham Oct 7 '18 at 12:36

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