Re: Categorical problems

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

On Mon, 8 Jun 2009, Eduardo J. Dubuc wrote:

> Ross Street wrote:
>
>> Problem. Suppose A is a locally small site whose category E of Set-
>> valued sheaves is also locally small. Is E a topos? (see (*) below)
>
> This is one of (probably) many problems of Girau topoi [satisfy all
> conditions
> in Girau's Theorem exept (may be) the set of generators] which are not known
> to be a topos.
>
> (*) It seems Not: Take a Girau (really faux but locally small) topos E, with
> the canonical topology. Then the topos of sheaves should be E again, which is
> not a topos (am I missing something ?).
>
I presume that Ross was using the word "topos" to mean "elementary topos".
But in any case, Eduardo was missing something: the proof that, if E is
an \infty-pretopos (my preferred name for what he calls a "Girau(d)
topos"), then every canonical sheaf on E is representable, requires the
existence of a generating set (see C2.2.7 in the Elephant). For a
counterexample in the absence of generators, let G be the "large" group
of all functions from the ordinals to {0,1} having finite support, the
group operation being pointwise addition mod 2 (or, if you prefer,
the group of finite subsets of the ordinals under symmetric difference),
and let E be the (elementary) topos of G-sets. For each ordinal \alpha,
let A_\alpha be the set {0,1} with G acting via its \alpha-th factor;
then any G-set admits morphisms into only a set of the A_\alpha, from
which it follows that the coproduct of all the A_\alpha exists as a
(set-valued) canonical sheaf on E, though it clearly isn't a set.
Moreover, this coproduct admits a proper class of maps to itself, so
the category of sheaves on E isn't locally small; hence it doesn't
violate Ross's conjecture.

Peter Johnstone



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