Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer 1

up vote 18 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).

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.