When is Fam(E) a topos?

When is Fam(E) a topos?

[Peter Johnstone]

Others may have noticed a slight gap in what I wrote on Saturday,
concerning the difference between toposes with set-indexed
copowers and those with coproducts. If E has copowers then the
functor Delta exists, but to prove that Fam(E) is equivalent to
the topos obtained by glueing along it you need arbitrary
coproducts. In fact these are necessary for Fam(E) to be
cartesian closed; I now have a proof of this, but it's a bit
too complicated to write out in ASCII. I plan to write it up as
a short paper.

Peter Johnstone


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when is Fam (E) a topos?

[Thomas Streicher]

A couple of days ago I made the wrong claim that

> If BB is a topos and P : XX -> BB is a fibration then P is a fibration
> of toposes iff XX is a topos and P is a logical functor.

The following shows how wrong this claim is.

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.

Thus, for the motivating examples of fibred toposes the total category
is a topos only in the trivial case!

Thomas


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Re: when is Fam (E) a topos?

[Thomas Streicher]

> 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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Re: when is Fam (E) a topos?

[Peter Johnstone]

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
>
>


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Re: when is Fam (E) a topos?

[Thomas Streicher]

Peter's argument also shows that a topos E has copowers of 1 already if
Fam(E) is cartesian closed. Thus, we have also examples of fibrations
of ccc's over a topos whose total category is not even cc.

Thomas


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