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Re-reading my own recently posted question What is the total space of a stack after all? I realized that I don't know something more simple and presumably more basic.

Are there (bounded if you like) geometric morphisms $f:\mathbf Y\to\mathbf X$ between (Grothendieck if you like) toposes with the following property: there is an object $X$ of $\mathbf X$ with global support (that is, $X\to1$ epi) such that the pullback $\mathbf Y/f^*X\to\mathbf X/X$ is equivalent, over $\mathbf X/X$, to the presheaf topos $(\mathbf X/X)^{{\mathbb C}^{\mathrm{op}}}\to\mathbf X/X$ for some internal category $\mathbb C$ in $\mathbf X/X$ but $f$ itself is not a presheaf topos over $\mathbf X$ for any internal category there?

If yes, what would be an illustrative example?

More generally, can it happen that $f$ is not bounded but the pullback is?

If yes, are there some (cohomological or otherwise) obstructions to globalizing local data of the above kind?

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    $\begingroup$ Essentially, no it cannot happen, at least in the case where everything is bounded. Essentially because of descent theory. Right now I'm lacking the time to write a full answer, but it follows from the same sort of argument as the two questions and answer by myself: mathoverflow.net/questions/159152/… mathoverflow.net/questions/233219/… roughly by replacing "coherent geometric morphism" by "relative presheaf topos" every where. $\endgroup$
    – Simon Henry
    Commented Aug 1, 2017 at 10:59
  • $\begingroup$ @SimonHenry What you say is very interesting. I don't know an abstract characterization of relative presheaf toposes. When the base is Sets, it is enough indecomposable projectives, but over more general bases I don't know. When you find time, I would be very grateful to learn this from you. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 1, 2017 at 13:48
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    $\begingroup$ To be clear I'm not saying that there is an internal characterization of internal presheaf topos other than the existence of an internal category such that etc... My point is more that if $Y/f^* x$ is presheaf over $X/x$ then the internal cauchy complete category over which it is a presheaf should comes with a descent data (maybe in the sense of anafunctors) because of its uniqueness, hence one should be able to show that this category is of the form $x \times \mathbb{C}$ for some some category in $X$, and use descent theory again to show that the isomorphism... $\endgroup$
    – Simon Henry
    Commented Aug 1, 2017 at 22:40
  • $\begingroup$ ... between $Y/f^*x$ and the category of $X/x$ valued Presheaves over $x \times \mathbb{C} $ descend into an isomorphism between $Y$ and the category of $X$ valued presheaves over $\mathbb{C}$ $\endgroup$
    – Simon Henry
    Commented Aug 1, 2017 at 22:42
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    $\begingroup$ I guess you are right in the general situation, but By "everything is bounded" I also meant that $X$ and $Y$ were Grothendieck toposes. Unless is correct the problems you mentione cannot happen for grothendieck toposes isn't it ? The stackification of a small pre-stack on a site is always small and the 2-category of internal categories (with anafunctors between them) is equivalent to the 2-category of stacks. $\endgroup$
    – Simon Henry
    Commented Aug 2, 2017 at 6:30

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