What conditions on a Grothendieck site $\left(C,J\right),$ are equivalent to the diagonal map $$Sh_J\left(C\right) \to Sh_J\left(C\right) \times Sh_J\left(C\right)$$ being a proper map of topoi?
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I think Johnstone's Elephant gives a site characterisation of proper maps between toposes; so I guess the problem reduces to finding the site corresponding to the product $Sh_J (C) \times Sh_J(C)$ - this is surely known?
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$\begingroup$ A completely random guess: is it the coproduct $C \coprod C$? $\endgroup$– David Roberts ♦Commented Apr 4, 2012 at 21:38
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$\begingroup$ No the coproduct of the site will yeld the categorical product of the categories of sheaf, wich is the co-product of the topos... If I'm not mistaken you obtain a site for the product by taking the product of the two site and constructing a suitable 'product topology' on it... $\endgroup$ Commented Apr 4, 2012 at 22:04