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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