Andre Joyal, M. Tierney, An extension of the Galois theory of Grothendieck Mem. Amer. Math. Soc. no 309 (1984)
An extension of the equivalence to morphisms is discussed in
Ieke Moerdijk, The classifying topos of a continuous groupoid I , Trans. Amer. Math. Soc. Volume 310, Number 2, (1988)
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.
Revised on October 23, 2011 15:30:27
by Dmitri Pavlov