In this post, I defined Grothendieck Topologies and gave some examples. Look there and in related links for more on Grothendieck topologies. We’re going to move on to Sheaves today. This is going to be a minipost, because this does deserve it’s own posting and it’s going to take me some time to work out the next in this series.

So we should start out by talking about what a sheaf is on a topological space. Let $X$ be a topological space, and recall that $\mathbf{Cat}_X$ is the category of open sets on $X$. Then a presheaf of objects of a category $\mathcal{C}$ is a contravariant functor $\mathbf{Cat}_X\to \mathcal{C}$. For simplicity, we will talk about presheaves of sets and of abelian groups, rather than the full generality.

A sheaf, then, is a presheaf with additional structure. Specifically we require that a sheaf $\mathscr{F}$ satisfy that if $U$ is any open set and $\{U_i\}$ is any open cover of $U$, and $U_{ij}$ is used to denote $U_i\cap U_j$, then if given $s_i\in \mathscr{F}(U_i)$ such that $s_i|_{U_{ij}}=s_j|_{U_{ij}}$ for all $i,j$, then there exists a unique $s\in \mathscr{F}(U)$ such that $s|_{U_i}=s_i$ for all $U_i$. That is, there exist unique gluings of local sections.

This is actually equivalent to the statement that, given an open set $U$ and an open cover $\{U_i\}$, then the sequence $\mathscr{F}(U)\to\prod_i\mathscr{F}(U_i)\rightrightarrows \prod_{i,j} \mathscr{F}(U_i\cap U_j)$ is exact. By that we mean that the first map has image the equalizer of the pair of maps to $\prod_{i,j}\mathscr{F}(U_i\cap U_j)$.

In light of this, given a site $\mathcal{C}$, a sheaf on $\mathcal{C}$ is a contravariant functor on $\mathcal{C}$ such that for every covering $\{U_i\to U\}$ the diagram $\mathscr{F}(U)\to\prod_i\mathscr{F}(U_i)\rightrightarrows \prod_{i,j}\mathscr{F}(U_i\times_U U_j)$ is exact.

Now, given a scheme $X$ over a scheme $S$, as we move through the various sites mentioned previously, it gets harder and harder to be a sheaf. The toughest topology for a functor to be a sheaf on is the fppf topology, which makes the fact that the functor $\hom_S(-,X)$ is an fppf sheaf. This functor is called the functor of points of $X$ over $S$.

Next in this series will be descent theory, I’m attempting to put it all together (which seems somehow fitting) and I’ll continue this line of thought when I am capable of doing so.