The answer to Mamuka's question (1) is no. Observe first that YY/f^*(X) is bounded over XX/X iff it's bounded over XX, since XX/X --> XX is bounded. And one has: Proposition: If Y has global support in YY and G is a bound for YY/Y over XX, then \Sigma_Y(G) is a bound for YY over XX. Proof: By assumption, any object B of YY/Y is a subquotient of some object G \times Y^*f^*(I), with I an object of XX. But the Frobenius reciprocity condition \Sigma_Y(G\times Y^*f^*(I)) \cong \Sigma_Y(G) \times f^*(I) holds, so \Sigma_Y(B) is a subquotient of \Sigma_Y(G)\times f^*(I). Finally, since Y has global support, any object A of XX is a quotient of \Sigma_Y(Y^*(A)) \cong A \times Y. I have no thoughts at present about question (2). Peter Johnstone On Mon, 31 Jul 2017, Mamuka Jibladze wrote:

Recently I posted this question

https://mathoverflow.net/q/277582/41291

to mathoverflow and now it occurred to me that most likely I can get a quick answer here.

Are there geometric morphisms f: YY -> XX which are

(1) locally but not globally bounded, or (2) locally but not globally presheaf, or (3) as in (2) and bounded?

In more detail, I mean this: there must be an object X in XX with global support (X->1 epic) such that the pullback f/X: YY/f^*(X) -> XX/X is

(1) bounded, while f is not bounded, or (2) equivalent over XX/X to the topos (XX/X)^{CC^op} of internal presheaves on some internal category CC of XX/X, while YY is not equivalent to any such over XX, or (3) same as (2) and in addition f bounded.

Can any of these happen?

Hoping, Mamuka

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