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.) (more…)