Sheaves as full reflective subcategories

Hello everyone.

My question is concerned with the following statement.

"Having a grothendieck topology on a category C is equivalent to having a full reflective subcategory Sh(C) in the category PSh(C) of presheaves, whose reflection is left exact."

What i need is a reference for this containing a proof. I tried google but could not find anything besides citations of this result.

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I have seen a reference for this fact, and I think it was in Artin's book on Grothendieck Topologies. I have no copy available to check this right now.

Before I found that reference, I wrote up a little treatment for my own benefit; I took the "full reflective subcategory" idea as the definition of a Grothendieck topos, then proved that all such come from Grothendieck topologies. It's in section 3.7 of http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf

The proof goes like this. That a Grothendieck topology gives rise to a full reflective subcategory with left-exact reflection is standard. If you're given such a reflective subcategory $D\subseteq Psh(C)$, consider all the sieves, i.e., monomorphisms $f:S\to h_X$ where $h_X$ is the representable functor determined by $X\in C$. Call $f$ a covering sieve if $Lf$ is an isomorphism, where $L: Psh(C)\to D$ is the left adjoint. You then show (i) the collection of covering sieves is a Grothendieck topology $\tau$, and (ii) sheaves for $\tau$ are exactly those presheaves isomorphic to objects of $D$. Both (i) and (ii) require using the fact that $L$ is left exact. (ii) is equivalent to the statement: (ii') for all $f:X\to Y$ in $Psh(C)$, $Lf$ is iso if and only if $L_\tau f$ is iso (where "$L_\tau$" is sheafification with respect to $\tau$.) It's convenient to prove (ii') first for monomorphisms $f$, and then for epimorphisms $f$.

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thank you very much. this is exactly what i needed. plus the rest of your notes seem interesting as well. – Garlef Wegart Feb 28 '10 at 16:20
just wanted to say, nice notes! – B. Bischof Mar 1 '10 at 2:33

To add to what Charles wrote, another reference is Mac Lane and Moerdijk's Sheaves in Geometry and Logic. They prove something a bit more general, involving Lawvere-Tierney topologies on a topos. For the purposes of understanding what I'm about to write, it's not necessary to know what a Lawvere-Tierney topology is.

Mac Lane and Moerdijk's book contains the following two results:

1. Let $\mathcal{E}$ be a topos. Then the subtoposes of $\mathcal{E}$ (i.e. the reflective full subcategories with left exact reflectors) correspond canonically to the Lawvere-Tierney topologies on $\mathcal{E}$.

2. Let $\mathbf{C}$ be a small category. Then the Lawvere-Tierney topologies on $\mathbf{Set}^{\mathbf{C}^{\mathrm{op}}}$ correspond canonically to the Grothendieck topologies on $\mathbf{C}$.

Result 1 is almost part of Corollary VII.4.7. The "almost" is because they don't go the whole way in proving the one-to-one correspondence, but I guess it's not too hard to finish it off. (Edit: it also appears as Theorem A.4.4.8 of Johnstone's Sketches of an Elephant, where Lawvere-Tierney topologies are called local operators.) Result 2 is Theorem V.4.1.

I agree with the point of view that Charles advocates. When I started learning topos theory I got bogged down in detailed stuff about Grothendieck topologies, and it all seemed pretty technical and unappealing. It wasn't until years later that I learned the wonderful fact that Charles mentions: an elementary topos is Grothendieck iff it's a subtopos of some presheaf topos. I wish someone had told me that in the first place!

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The mentioned references and some more are at nLab: category of sheaves.

For instance the book by Kashiwara-Shapira has a useful account. When I myself learned this stuff I found it useful to read Mac-Lane/Moerdijk in parallel to Kashiwara/Shapira. The former has more of the topos-theoretic picture, the latter more of the homotopy-theoretic picture.

As we know from Rezk and Lurie, it*s both these aspects taken together that give the full picture.

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