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.



Sheaves as full reflective subcategories
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 $f$.


