Le 2016-11-08 11:03, Thomas Streicher a ??crit??:
On Mon, Nov 07, 2016 at 04:03:18PM -0500, Eduardo Julio Dubuc wrote:
Hi, in this posting I will use the terminology used by most people in the list.
There are Grothendieck, Giraud and Elementary (Lawvere-Tierney) topos.
Grothendieck are Giraud and Elementary, my question is:
Are Elementary Giraud topos which are not Grothendieck ?
But Grothendieck and Giraud toposes are the same. In the Elephant one can even find a relative Giraud Theorem.
Thomas
Grothendieck toposes bear a similar relationship to elementary toposes as Grothendieck abelian categories to abelian categories. Recall that a Grothendieck abelian category is an abelian category which (1) has filtered colimits cummuting with finite limits (this is Grothendieck's condition AB5), (2) has a set of generators. Any Grothendieck topos is a (locally small) elementary topos which satisfies (1) and (2) because it is a localization of a category of presheaves on a small category. Conversely, any (locally small) elementary topos satisfying (1) and (2) satisfies the 6 axioms of Giraud, and is thus a Grothendieck topos. Indeed, the only two axioms of Giraud not satisfied in a general elementary topos are the existence of small (disjoint and universal) coproducts which for an elementary topos follows from (1), and the existence of a set of generators which is (2). All the best, Clemens. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]