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Can we obtain the bicategory of Grothendieck toposes and geometric morphisms as a bicategorical localization?
A statement that I have in mind is that the bicategory of Grothendieck toposes and geometric morphisms should be something like the bicategorical localization of the 1-category of localic groupoids, (internal) functors, and some version of (internal) essentially surjective fully faithful functors as weak equivalences.
Has anything like this appeared in the literature?
I am aware of the results mentioned at classifying topos of a localic groupoid, where Joyal–Tierney and Moerdijk get us pretty close to such a statement, but not quite.
The original work of Pronk seems to treat only etale groupoids and etendues.
There was discussion of this on the category theory Zulip chat recently. I think the construction of Moerdijk of a localic category from a localic groupoid, where he then shows their categories of sheaves are equivalent, is something that might be able to be avoided, if we take localic categories rather than localic groupoids as the basic input. If we want the non-invertible 2-arrows, then localic groupoids and the functors between them is not sufficient. Moerdijk’s notion of morphism between localic groupoids in the localisation looks like bitorsors between the associated localic categories.
I definitely think it should be possible, and I don’t think it’s in the literature.
My suspicion is that it should help to consider localic groupoids as a category enriched over double categories, where one direction of the hom-double-categories consists of internal natural transformations in the category of locales and the other direction consists of levelwise inequalities with respect to the locally-ordered nature of the category of locales. Neither of these two kinds of 2-cells alone can possibly carry all the information in the 2-cells of toposes, since the former is invertible while the latter is thin, but when combined there is some hope.
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