## Sheaves

Alright so, with all the algebraic geometry lined up for the Spring and Fall, we might as well get a head start. That’s right — time to define sheaves. I admit these things sound pretty intimidating at first, but I think I see why you might want to talk about them. The idea is that you have some topological space $X$ and you want to attach some algebraic data to the open sets. For example, you might want to attach to each open set $U$ of $X$ the ring $C(U)$ of continuous functions $U \to \mathbb{C}$, say. Or, if you were working with a variety, you could talk about the regular functions on $U$ instead.

Here is the definition (from Hartshorne). A presheaf $\mathcal{F}$ of abelian groups on $X$ consists of the data

• for every open subset $U \subseteq X$, an abelian group $\mathcal{F}(U)$, and
• for every inclusion $V \subseteq U$ of open subsets of $X$, a morphism of abelian groups $\rho_{UV} : \mathcal{F}(U) \to \mathcal{F}(V)$

subject to the conditions

• $\mathcal{F}(\varnothing) = 0$, where $\varnothing$ is the empty set,
• $\rho_{UU}$ is the identity map $\mathcal{F}(U) \to \mathcal{F}(U)$, and
• if $W \subseteq V \subseteq U$ are three open subsets, then $\rho_{UW} = \rho_{VW} \circ \rho_{UV}$.

That’s quite a mouthful, especially considering we’re defining something which isn’t even good enough to be a legit sheaf. However, you probably see what’s going on. The first two points may have had you thinking: “$\mathcal{F}$ is a functor”. Once you read the last three points, though, your suspicions were surely confirmed.

Indeed, the above merely defines a presheaf $\mathcal{F}$ to be a (contravariant) functor $\mathcal{O}(X) \to \mathbf{Ab}$ where $\mathbf{Ab}$ is the category of abelian groups. Here, $\mathcal{O}(X)$ is a category “manufactured out of $X$“. Its objects are the open sets of $X$, and the morphisms are the inclusions. Simple as that. One can easily define presheaves of other algebraic structures than abelian groups, just by replacing the target category.

For $U$ an open set, we call $\mathcal{F}(U)$ the section of the presheaf $\mathcal{F}$ over $U$. The maps $\rho_{UV}$ are termed restriction maps and we sometimes write $\left. s \right|_V$ rather than $\rho_{UV}(s)$ if $s \in \mathcal{F}(U)$.

While presheaves may seem like a reasonably good tool for accomplishing the task of “attaching algebraic data to open sets” that we first mentioned, they turn out to be a bit too general. We need to ask a bit more of these functors to ensure they behave properly in the contexts we wish to employ them.

There are two more assumptions we need to impose on presheaves before they qualify as sheaves:

• If an open set $U$ is covered by open sets $\{ V_i \}$ and $s \in \mathcal{F}(U)$ is such that $\left. s \right|_{V_i} = 0$ for all $i$, then indeed $s=0$.
• If an open set $U$ is covered by open sets $\{ V_i \}$ and we have a bunch of “pieces” $s_i \in \mathcal{F}(V_i)$ for each $i$ which “agree” on the overlaps in the sense that $\left. s_i \right|_{V_i \cap V_j} = \left. s_j \right|_{V_i \cap V_j}$ then there is a way of “gluing them together”: there is some $s \in \mathcal{F}(U)$ such that $\left. s \right|_{V_i} = s_i$ for each $i$.

Next time we will discuss some pathological examples which assure us that indeed, both of these additional axioms are required. After that we’ll talk about morphisms of sheaves. As you probably guessed, those will be natural transformations…