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One of my friends asked me whether or not the inclusion of the category of Grothendieck toposes into elementary toposes has a left adjoint. We are looking at the categories of geometric morphisms. I am not really sure how to start but nothing seems to rule it out immediately.

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up vote 19 down vote accepted

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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Any time you find yourself wondering “does such-and-such functor have an adjoint?”, checking this sort of condition (preservation of (co)limits in general, and of simple ones like initial/terminal objects in particular) is the first thing to try. – Peter LeFanu Lumsdaine Dec 18 '10 at 22:29

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