RE: stacks (was: size_question_encore)

[Eduardo Dubuc <edubuc@dm.uba.ar>]

Dear all:

If we take the filtered poset of finite subsets of I, then taking the 
finite products of C(2) gives an strict pro group, and this is the 
example of a "Faux topos", SGA4 SLN 269 page 322.

Now, this example is exhibited as what we some times call Giraud topos 
(all exactness properties but without generators)

My question is (answer probably well known to the experts):

Are Giraud topoi elementary topoi (that is, do they have an Omega and 
exponentials) ?

greetings   e.d.

On 07/12/2011 12:04 PM, André Joyal wrote:
> Dear Michael,
>
> You wrote:
>
> > Are there known examples of elementary toposes which violate the
> axiomof stack completions?
>
> Here is my favorite example.
>
> Let C(2) be the cyclic group of order 2.
> It suffices to construct a topos E for which the
> cardinality of set of isomorphism classes of C(2)-torsor is larger
> than the cardinality of the set of global sections of any object of E.
>
> Let G=C(2)^I be the product of I copies of C(2), where I is an infinite
> set.
> The group G is compact totally disconnected.
> Let me denote the topos of continuous G-sets by BG.
>
> There is then a canonical bijection between the following three sets
>
> 1) the set of isomorphism classes of C(2)-torsors in BG
>
> 2) the set of isomorphism classes of geometric morphisms BC(2)--->BG
>
> 3) the set of continuous homomomorphisms G-->C(2).
>
> Each projection G-->C(2) is a continuous homomomorphism.
> Hence the cardinality of set of isomorphism classes of C(2)-torsors in BG
> must be as large as the cardinality of I.
>
> The topos E=BG is thus an example when I is a proper class.
>
> For those who dont like proper classes, we may
> and take for E the topos of continuous G-sets in a
> Grothendieck universe and I to be a set larger than this universe.
>
> Best,
> Andre

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