Re: Giraud_Elementary_?

[henry@phare.normalesup.org]

Dear Eduardo

Unless you are using a different statement of the Giraud's theorem than
the one I have in mind, they are I think considerably more often called
$\infty$-pretopos (like in the Elephant) or infinitary pretopos (like in
the nLab) to avoid any confusion with an infinity categorical notion. I
don't think I have ever encountered a different terminology (but I do like
'Giraud topos').

Regarding the example you are looking for, unless I'm missing something,
the example 2.8 in SGA that you mentioned (the category sets endowed with
smooth action of a large group) is also an elementary topos:  sub-object
classifier, exponential and power object are constructed exactly in the
case of an ordinary group action topos and only involve a small quotient
of the large group. So it answer you question.

Bests,
Simon

> By Giraud topos I mean all the assumptions in Giraud's theorem, exept a
> small set of generators. What Grothendieck call "faux topos".
>
> See SGA4 Exposse IV
> Theoreme 1.2 (Giraud's theorem) and Example 2.8 (faux topos).
>
> best  e.d.
>
> I guess I was wrong when I thought that "Giraud Topos" was established
> terminology in the cat-list.
>
>

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