The site of G-sets and the associated category of sheaves

In the past, I said a few words about Grothendieck topologies and fpqc descent. Well, strictly speaking, I didn’t get very far into the descent bit. I described a topology on the category of schemes (the fpqc topology) and showed that it was a subcanonical topology, that is, any representable presheaf was a sheaf in this topology.

This amounted to saying that if ${X' \rightarrow X}$ was a fpqc morphism of schemes, then to hom out of ${X}$ was the same thing as homming out of ${X'}$ such that the two pull-backs to ${X' \times_X X'}$ were the same. If I had gotten further, I would have shown that to give a quasi-coherent sheaf on ${X}$ (among other things) is the same as giving “descent data” of a quasi-coherent sheaf on ${X'}$ together with an isomorphism between the two pull-backs to ${X' \times_X X'}$ satisfying the cocycle condition. Maybe I’ll do that later. But there is a more basic “toy” example that I now want to describe of a site (that is, category with a Grothendieck topology) and the associated category of sheaves on it.

1. ${G}$ -sets

Our category ${\mathcal{C}}$ is going to be the category of left ${G}$ -sets for a fixed group ${G}$ ; morphisms will be equivariant morphisms of ${G}$ -sets. We are now going to define a Grothendieck topology on this category. For this, we need to axiomatize the notion of “cover.” We can do this very simply: a collection of maps ${\left\{U_i \rightarrow U\right\}}$ is called a cover if the images cover ${U}$ . Now, fiber products of ${G}$ -sets are calculated in the category of sets, or in other words the forgetful functor

$\displaystyle G-\mathbf{set} \rightarrow \mathbf{Sets}$

commutes with limits (as it has an adjoint, the functor ${S \mapsto G \times S}$ ). Thus, taking pull-backs preserve the notion of covering, and it is easy to see the other axioms are satisfied too: if we have a cover of each of the ${U_i}$ (which cover ${U}$ ), then collecting them gives a cover of ${U}$ . Similarly, an isomorphism is a cover. This is obvious from the definitions.

2. Representable presheaves

So we indeed do have a perfectly good site. Now, we want a characterization of all the sheaves of sets on it. To start with, let us show that any representable functor forms a sheaf; that is, the topology is subcanonical. (In fact, this topology is the canonical topology, in that it is the finest possible that makes representable functors into sheaves.)

Proposition 1 Any representable functor on ${G-\mathbf{set}}$ is a sheaf in the above topology.

To see this, we have to show the following fact (in view of the fact that . If ${X' \rightarrow X}$ is a surjection of ${G}$ -sets and we have a map ${X' \rightarrow R}$ (for some other ${G}$ -set ${R}$ ) such that the two pull-backs

$\displaystyle X' \times_X X' \rightarrow X'$

become equal, then we get a map ${X \rightarrow R}$ . But if we are thinking of maps of sets, then any two points in ${X' }$ that map to the same thing in ${X}$ clearly go to the same place in ${R}$ by the above condition. So we can get a sequence

$\displaystyle X' \twoheadrightarrow X \rightarrow R$

where the composite is the given map ${X' \rightarrow R}$ . We only need to see that ${X \rightarrow R}$ is a homomorphism. But this is clear since ${X' \rightarrow R}$ is and ${X' \rightarrow X}$ is a surjective homomorphism.

3. The classification of sheaves

Now that we have shown that representable presheaves on ${G-\mathbf{set}}$ are sheaves, we want to get the converse direction. We want to show that every sheaf is representable. In fact, we will show that there is a correspondence between sheaves and ${G}$ -sets.

Theorem 2 There is an equivalence of categories $\displaystyle \left\{\text{sheaves on } G-\mathbf{set}\right\} \simeq G-set$

that assigns to each ${G}$ -set ${R}$ the sheaf ${X \mapsto \hom_G(X, R)}$ .

In fact, we are going to define the inverse functor. Given a sheaf ${\mathcal{F}}$ , we consider ${\mathcal{F}(G)}$ . Since ${G}$ acts on ${G}$ on the right by morphisms of left ${G}$ -sets, ${\mathcal{F}(G)}$ is naturally a (left!) ${G}$ -set.

We are going to show that these two functors are inverse to each other. One direction is straightforward. If ${R}$ is a ${G}$ -set, then the ${G}$ -set of maps of left- ${G}$ sets ${G \rightarrow R}$ is canonically isomorphic to ${R}$ ; if we have a map ${\phi: G \rightarrow R}$ , we send it to ${\phi(1)}$ . It is easy to check that for each element of ${R}$ , we do indeed get such a map ${\phi}$ , and that the left- ${G}$ -structure on ${\hom_G(G, R)}$ is the same as that on ${R}$ .

For the other, let ${\mathcal{F}}$ be a sheaf on ${G-\mathbf{set}}$ . We want to construct a natural isomorphism

$\displaystyle \mathcal{F}(X) \simeq \hom_G(X, \mathcal{F}(G)).$

It is clear that we have a map of ${G}$ -sets from ${G \times |X|}$ (where ${G \times |X|}$ denotes the ${G}$ -set ${G \times X}$ but with ${G}$ only acting on the first factor) into ${X}$ . In fact, we have a surjection of ${G}$ -sets

$\displaystyle G \times |X| \twoheadrightarrow X.$

There is thus an exact (equalizer) sequence by sheafiness

$\displaystyle \mathcal{F}(X) \rightarrow \prod_X \mathcal{F}(G) \rightrightarrows \prod_{T} \mathcal{F}( G)$

where ${T}$ is some set. We have used the fact that the fibered product of two copies of ${G}$ is always ${G}$ or ${\emptyset}$ . From this, it is clear that there is a natural injective map from ${\mathcal{F}(X)}$ into the set of functions ${X \rightarrow \mathcal{F}(G)}$ . These functions ${X \rightarrow \mathcal{F}(G)}$ are in fact ${G}$ -equivariant because of the way ${G \times |X| \rightarrow X}$ was defined. Now we want to check that the map is surjective.

For this, let us use a version of the “finite presentation trick.” We have seen that that for any ${G}$ -set ${X}$ , there is a coequalizer diagram

$\displaystyle F' \rightrightarrows F \twoheadrightarrow X$

where ${F', F}$ are free ${G}$ -sets. Applying ${\mathcal{F}}$ and ${\hom_G(-, \mathcal{F}(G))}$ turns this into two equalizer diagrams of sets with a morphism

To see that both are equalizer diagrams, we use the fact that ${\mathcal{F}}$ is a sheaf and representable functors are sheaves. It is clear that the two rightmost vertical arrows are isomorphisms, so the leftmost one is an isomorphism too. This finishes the proof.

From this, we see that giving a sheaf on ${G}$ –sets equates to giving a ${G}$ -set. Giving a sheaf of abelian groups on this site thus equates to giving an abelian group with a compatible action of ${G}$ on it, i.e. a ${G}$ -module. With this in mind, we are going to get a toy example of sheaf cohomology on a site—namely, group cohomology.

I learned this from the book by Tamme on Etale cohomology. I’m reading it, and I’m repeatedly struck at how clear and accessible the exposition is there (especially compared with other treatments like SGA, which I find somewhat intimidating).

Climbing Mount Bourbaki