Author: Dmitri Pavlov Format: TextI added the following remark to [[classifying topos of a localic groupoid]].
It would be nice if somebody more competent in this area expanded it.
The above equivalence of categories can in fact be lifted to an equivalence between the bicategory of localic groupoids, complete flat bispaces, and their morphisms and the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations. The equivalence is implemented by the classifying topos functor, as explained in
Ieke Moerdijk, The classifying topos of a continuous groupoid II,
Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168.
The above equivalence of categories can in fact be lifted to an equivalence between the bicategory of localic groupoids, complete flat bispaces, and their morphisms and the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations. The equivalence is implemented by the classifying topos functor, as explained in
Ieke Moerdijk, The classifying topos of a continuous groupoid II, Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168.