Given a Grothendieck topos, what does its localic groupoid look like?  
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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 
 A: DISCLAIMER: This answer does not provide references,  nor is a formally thought-out proof. 
Only some (hopefully) useful heuristics.
You begin with a topos E with associated theory T(E). 
From T one creates a locale L (and a localic groupoid G(L) ), such that there is a surjection Sh(L)----> E, where Sh(L) is the topos of ALL sheaves on L. The topos E is then recovered by isolating the equivariant sheaves, ie G(L) -equivariants.  Now, let T(Sh(L)) the theory of the topos Sh(L). 
The question is basically: how T(sh(L)) relates to the original T(E). 
Conjecture: there is some kind of modality on T(sh(L)) such that T(E) is gotten as the "fixed points" for that modality. 
The intuition behind this conjecture is that the topos E is obtained by considering only well-behaved sheaves in the pool of all sheaves. They are, in a sense, the ones which are invariant with respect to some shuffling of the topos sh(L) by some suitable action. 
NOTE: My (unpublished) dissertation had a  somewhat germane theme: what I was after was some "toposophical" semantics of  general modal logics, and the trick was to consider topoi endowed with an extra lex endofunctor. The endofunctor was then used to isolate a fixed-points subcat (or more generally a cat of coalgebras for the endo) which had the nice property that the subobject functor had a built-in modal operator (induced by the endo). I do not know if the Joyal -Tierney representation produces something along similar lines, but the guess is yes. 
A: Starting from a topos $T$, construct a locale $L$ and a surjection $L \to T$ 'nice enough' (like a proper surjection). Then $(L, L \times TL,L \times TL \times TL)$ is a truncated simplicial locale, which can be seen as a localic groupoid. There is a canonical geometric morphism from the topos of sheaves on this groupoid to $T$, and if the surjection $L \to T$ was nice enough it's an isomorphism.
