Monday, 2 April 2007

topos theory - Is there a "Grothendieckification" functor from elementary toposes to Grothendieck toposes?

No, it doesn't. If it did, then it would preserve limits. But the category of Grothendieck toposes and geometric morphisms has a terminal object, namely the category of sets, while there are elementary toposes not admitting any geometric morphism to Set (for instance, any small elementary topos).

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