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I wish we had a decent account at sheaf toposes are the accessible left exact reflective subcategories of presheaf toposes.
Presently the account that the $n$Lab gives of the important fact is not very good. It’s stated at category of sheaves with proof by pointer to reflective (infinity,1)-category, which in turn points to Lurie’s HTT. The crucial point about the accessibility condition is presently discussed, very briefly, without cross-links at reflective subcategory here.
So all this should be collected coherently in one place, which we can then link to.
I seem to remember that the statement was a bit scattered in the Elephant, and I thought we had pointers to the relevant propositions in the Elephant on the $n$Lab, but now I don’t find them anymore.
Ah, it’s explicit in Borceux, Handbook of Categorical Algebra vol. 3 corollary 3.5.5, using the definition of “localization = lex reflection” from vol 1, def.3.5.5.
Hm, no mentioning of accessibility. I am confused about this point.
For 1-toposes, accessibility is automatic.
Okay, thanks. How does that follow?
Not only are sheaf toposes locally presentable, they are cototal categories; see here. In that situation the following version of the adjoint functor theorem applies: a functor $i: E \to F$ between sheaf toposes has a left adjoint iff it preserves limits. Now apply Theorem 2.2 of adjoint functor theorem.
What fails in the $\infty$-case, though (at least apparently — I don’t know a counterexample), is that a left exact localization of a Grothendieck $(\infty,1)$-topos may no longer be a Grothendieck $(\infty,1)$-topos. So I would turn your argument around, Todd: the reason that the subtopos is itself Grothendieck is because the localization functor is accessible, not the other way around. Morally, the localization functor is accessible because it’s determined by a universal closure operator and hence by an endomorphism of the impredicative subobject classifier, which is a “small amount of data”. More generally, any topological localization of an $(\infty,1)$-topos is automatically accessible, for essentially the same reason; see HTT 6.2.1.5.
Well, for any onlookers: nothing I said is incorrect, and it does provide an answer to Urs’s question.
Or to be more precise, I thought the question was: why is an inclusion $i: E \to F$ (a fully faithful functor with a lex reflector) between sheaf toposes accessible? In that case, you don’t need to prove that $E$ is Grothendieck; it’s Grothendieck because sheaf toposes are Grothendieck toposes by definition. And that was the question I was answering.
Now maybe you’re addressing Urs’s actual underlying concern more directly (which has an eye toward the $(\infty, 1)$-situation). In which case, fine. Of course, the argument I’m offering up has a different kind of generality to it.
Thanks, Todd and Mike.
Sorry, my question wasn’t good, it is answered already by just the adjoint functor theorem for locally presentable categories. Thanks to both of you for answering one of two better questions that I should have asked instead! :-)
I have made a minimum note at sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes, not, however, yet incorporating either of what you said.
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