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Stack theoretic interpretation of localic groupoid presentations

Adam Dauser
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Prof. Caramello's and her PhD students' talks inspired me to write up the following, it is modeled on the definition of algebraic stacks in algebraic geometry.

We seek to understand topoi as a certain class of stacks on the big site Sp of locales with étale covers (if you restrict to topoi with enough points, you can take sober topological spaces), that are of the form  X\mapsto \text{Core(Geom(Sh(}X),\mathcal{E})) for a topos  \mathcal{E} . Here Core denotes the maximal subgroupoid of a category.

The first insight (which was essentially mentioned by Prof. Caramello in her talk) is that by weak left Kan extending the functor \text{Sh}:Sp\to \text{Core}_{(2,1)} Topos along the Yoneda embedding y:Sp\to Stacks(Sp), we get a (2,1)-adjunction ( \text{Core}_{(2,1)} is the maximal sub-(2,1)-category):
 Lan_y\text{Sh}:\text{Core}_{(2,1)} Topos\to Stacks(Sp)
y:Stacks(Sp)\to \text{Core}_{(2,1)} Topos
defined by
 X\mapsto \text{Core(Geom(Sh(}X),\mathcal{E})) .
The idea now is to characterize topoi as the quotient stacks of localic groupoids: We claim a stack \mathcal{X} on Sp is in the essential image of y iff:

  1. The diagonal \Delta:\mathcal{X}\to \mathcal{X}\times \mathcal{X} is representable i.e. for any representable functor yX for a locale X and map f:yX\to \mathcal{X}\times \mathcal{X}, the pullback along \Delta is again representable.
    This corresponds to the fact, that the diagonal morphism of a topos \mathcal{E} is a localic geometric morphism. Thus, the base change of the diagonal along any map \text{Sh}(X)\to \mathcal{E}\times \mathcal{E} will be a localic geometric morphism with target \text{Sh}(X) and hence a localic topos itself.
  2. There exists a locale X and a map p:yX\to\mathcal{X} with the following property: For any locale Y and map f:yY\to\mathcal{X}, the base-change (which is representable by a locale Y' which follows from (i)) \bar{p}: yY'\to yY corresponds to an open surjection of locales Y'\to Y. This corresponds to the fact, that any topos admits an open surjective geometric morphism from a localic one.

From this data, we can reconstruct a localic groupoid and hence a topos.
This presentation would be more convenient than the localic groupoids themselves, because they tend to be rather big and complicated. For the same reason, algebraic stacks instead of "smooth groupoids in algebraic spaces" are used in algebraic geometry.

My question is now, whether this is a well-known characterization. And if not, whether this is actually correct and if somebody wants to help me with working out the details.

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