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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.

edited May 22 2010 at 8:47

Kevin Lin
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asked May 21 2010 at 12:49

Steven Gubkin
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Mike Shulman · Accepted Answer · 2010-05-21 18:27:18Z UTC

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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).

answered May 21 2010 at 18:27

Mike Shulman
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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 2010 at 22:29

Is there a “Grothendieckification” functor from elementary toposes to Grothendieck toposes?

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