Re: when is Fam (E) a topos?

[Peter Johnstone <ptj@dpmms.cam.ac.uk>]

I was surprised by Thomas's previous post, because I knew that
if E has set-indexed copowers then Fam(E) can be identified with
the glueing of Delta, and is thus a topos. (I haven't seen Pieter
Hofstra's thesis, so I wasn't aware that he had made a different
claim.)

In fact set-indexed copowers in E (a slightly weaker condition
than cocompleteness, cf. A2.1.7 in the Elephant) is necessary as
well as sufficient for Fam(E) to be a topos. Here's a proof:

Let me write objects of Fam(E) in the form (I, (A_i | i \in I))
where I is a set and the A_i are objects of E. Noting that the
forgetful functor  sending (I,(A_i)) to I is represented by the
object (1,(0)) where 0 is the initial object of E, it's easy to
see that if Fam(E) is cartesian closed then objects of the form
(1,(A)) form an exponential ideal, i.e. any exponential
(1,(A))^(I,(B_i)) is of the form (1,(C)). In particular, if the
exponential (1,(A))^(I,(1 | i \in I)) exists, it is of the form
(1,(C)) where C is an I-fold power of A in E. So E has arbitrary
set-indexed powers; but E^op is monadic over E, so it also has
set-indexed powers, i.e. E has set-indexed copowers.

Peter Johnstone

On Fri, 21 Apr 2017, Thomas Streicher wrote:

>> 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
>
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]