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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

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This is a duplicate of my question mathoverflow.net/questions/84197/… –  Benjamin Steinberg Jul 19 '12 at 16:52
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marked as duplicate by Benjamin Steinberg, S. Carnahan Jul 20 '12 at 3:47

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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.

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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NOTE: My (unpublished) dissertation had a somewhat germane theme: what I was after was some "toposophical" semantics of general modal logics, and the trick was to consider topoi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.

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