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Suppose $\mathbb{T}$ is a geometric theory, $\mathcal{E}$ is a topos, and $M$ is a model of $\mathbb{T}$ in $\mathcal{E}$. Is there any sort of elementary condition on $M$ and $\mathcal{E}$ (or, even better, on the geometric morphism $\mathcal{E}\to \mathbf{Set}$) which would allow us to recognize $\mathcal{E}$ as the classifying topos of $\mathbb{T}$ and $M$ as the generic $\mathbb{T}$-model therein?

I feel like this is a long shot, but I thought I would ask anyway.

Edit: Of course, such a condition could not be expressed in the internal logic of $\mathcal{E}$ (even including non-geometric logic), since then it would be preserved in all slices $\mathcal{E}/X$. This is one reason I feel it's a long shot; but the example of principal bundles mentioned in the comments suggests that it's not an entirely unreasonable question.

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Does anyone even know such a condition for the special case when this question is equivalent to asking "if we're given a principal G-bundle on a space X, can we tell if X = BG and the bundle is the canonical one?" ? –  Dylan Wilson Dec 21 '11 at 5:30
    
(that question wasn't rhetorical- I'd be interested to know if there's an answer!) –  Dylan Wilson Dec 21 '11 at 5:31
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Dylan, if the total space of the bundle is weakly contractible. See for instance theorem 7.4 here: www-math.mit.edu/~mbehrens/18.906/prin.pdf –  Urs Schreiber Dec 21 '11 at 9:04
    
Silly me! Now I somehow have more faith in Mike's question being answered... –  Dylan Wilson Dec 21 '11 at 9:15
    
Too hard question for me, but I see that Olivia Caramello has worked so much (and still working about) classyfing topoi and theories. –  Buschi Sergio Dec 21 '11 at 19:36

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