Toposes (topoi) as classifying toposes of groupoids
For example, if a topos E is the object classifier, or the preseaf topos on a small category C, is there a way of describing its localic groupoid? More generally, is there a way of describing the localic groupoid of the classifying topos of a geometric theory T in terms of T?
(By 'the localic groupoid of a Grothendieck topos E' I mean the localic groupoid, G, such that E = BG, where BG is the category G equivariant sheaves - such a thing is not uniquely determined, so I'll take any description.)
Has someone written down this dictionary somewhere already?
Thank you, Christopher