Re: when is Fam (E) a topos?

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

> Let E be a topos then Fam(E) -> Set is certainly a fibered topos
> but by Th.6.2.3 of Pieter Hofstra's Thesis Fam(E) is a topos iff E is
> an atomic category (in the sense of Johnstone's 1977 book on Topos Theory,
> exercise 12 on p. 257). But in atomic categories all morphisms are epic
> and thus Fam(E) is a topos only if E is trivial.

Alas, there is a flaw in Pieter's Th.6.2.3 (which certainly is not
crucial for the main results of his otherwise very nice Thesis).
Actually, it can be seen quite easily: if E is a cocomplete topos then
Fam(E) is equivalent to the glueing of Delta : Set -> E which is known
to be a topos.

So it seems to be open to characterize those toposes E for which
Fam(E) is a topos. In particular, I don't know the answer for E the
free topos (with nno) or a realizability topos. In the latter case we
know that glueing of Nabla (right adjoint to Gamma) is a topos but
it's different from Fam(E).

I'd be grateful about any suggestions even for these particular cases!

Thomas


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