Sheaf

Definition

Let $X$ be a topological space. A presheaf $\mathcal F$ of sets on $X$ consists of the following data. For each open set $U \subseteq X$ there is a set $\mathcal F(U)$. For each pair of open sets $V \subseteq U$ there is a map $\rho_{UV} \colon \mathcal F(U) \to \mathcal F(V)$, written $s \mapsto s|_V$. These satisfy $\rho_{UU} = \id$ and $\rho_{VW} \circ \rho_{UV} = \rho_{UW}$ whenever $W \subseteq V \subseteq U$. The elements of $\mathcal F(U)$ are the sections of $\mathcal F$ over $U$. A presheaf $\mathcal F$ is a sheaf if, for every open set $U$ and every open cover $U = \bigcup_{i \in I} U_i$:

1. (Locality) if $s, t \in \mathcal F(U)$ satisfy $s|_{U_i} = t|_{U_i}$ for all $i$, then $s = t$;
2. (Gluing) if $s_i \in \mathcal F(U_i)$ satisfy $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ for all $i, j$, then there is $s \in \mathcal F(U)$ with $s|_{U_i} = s_i$ for all $i$.

A sheaf of abelian groups is a sheaf in which each $\mathcal F(U)$ is an abelian group and each restriction map is a homomorphism. A sheaf of commutative rings is defined in the same way.