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

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.

share|improve this question

3 Answers 3

up vote 10 down vote accepted

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 $f$.

share|improve this answer
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!

share|improve this answer

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.

share|improve this answer

Your Answer


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.