The Stacks project

Definition 6.30.2. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$.

1. A sheaf $\mathcal{F}$ of sets on $\mathcal{B}$ is a presheaf of sets on $\mathcal{B}$ which satisfies the following additional property: Given any $U \in \mathcal{B}$, and any covering $U = \bigcup _{i \in I} U_ i$ with $U_ i \in \mathcal{B}$, and any coverings $U_ i \cap U_ j = \bigcup _{k \in I_{ij}} U_{ijk}$ with $U_{ijk} \in \mathcal{B}$ the sheaf condition holds:

   • For any collection of sections $s_ i \in \mathcal{F}(U_ i)$, $i \in I$ such that $\forall i, j\in I$, $\forall k\in I_{ij}$

     \[ s_ i|_{U_{ijk}} = s_ j|_{U_{ijk}} \]

     there exists a unique section $s \in \mathcal{F}(U)$ such that $s_ i = s|_{U_ i}$ for all $i \in I$.

2. A morphism of sheaves of sets on $\mathcal{B}$ is simply a morphism of presheaves of sets.