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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


marked as duplicate by Benjamin Steinberg, S. Carnahan Jul 20 '12 at 3:47

This question was marked as an exact duplicate of an existing question.


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NOTE: My (unpublished) dissertation had a somewhat germane theme: what I was after was some "toposophical" semantics of general modal logics, and the trick was to consider topoi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


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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