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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.
For the case of invertible 2-morphisms, this is asserted (but unfortunately proved only for homotopy 1-categories) in Ieke Moerdijk’s The classifying topos of a continuous groupoid I, Theorem 7.7. Specifically, bicategorically localizing etale-complete localic groupoids (with open source and target maps) at open essentially surjective fully faithful functors produces the bicategory of Grothendieck toposes, geometric morphisms, and natural isomorphisms.
Part II deals with noninvertible 2-morphisms, but Moerdijk only formulates it as an equivalence of the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations (not necessarily invertible) and the bicategory of localic groupoids (with open source and target maps), complete flat bibundles between their completions, and homomorphisms of bibundles. However, Part II does not say anything about bicategories of fractions.
Re #3:
There was discussion of this on the category theory Zulip chat recently.
Is there a way to see the discussion without registering? (Is there any particular reason why the chat not available for public viewing?)
There was meant to be a public archive/readable version, but I don’t know what happened with that yet. I’m messaged one of the mods.
Is there a citeable reference for the fact that taking the bicategory of fractions of localic etale groupoids with respect to fully faithful essentially surjective functors produces a bicategory equivalent to the bicategory of etendues with invertible 2-morphisms, where etendues are understood in the generalized localic sense?
Something like Corollary 35 in Pronk’s Etendues and stacks as bicategories of fractions, but for locales instead of topological spaces?
I suspect not, though one might be able to cobble it together using something of mine for the localic groupoid side (to get a bicategory of fractions), and then using the comparison theorem Pronk proves, pointing out any changes (or not) necessary from the topological etendue version.
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