Do there exist nontrivial locally bounded geometric morphisms and/or locally (pre)sheaf toposes?

Do there exist nontrivial locally bounded geometric morphisms and/or locally (pre)sheaf toposes?

[Mamuka Jibladze]

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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Re: Do there exist nontrivial locally bounded geometric morphisms and/or locally (pre)sheaf toposes?

[Peter Johnstone]

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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