The Stacks project

Definition 6.7.1. Let $X$ be a topological space.

1. A sheaf $\mathcal{F}$ of sets on $X$ is a presheaf of sets which satisfies the following additional property: Given any open covering $U = \bigcup _{i \in I} U_ i$ and any collection of sections $s_ i \in \mathcal{F}(U_ i)$, $i \in I$ such that $\forall i, j\in I$

   $$s_ i|_{U_ i \cap U_ j} = s_ j|_{U_ i \cap U_ j}$$

   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 is simply a morphism of presheaves of sets.

3. The category of sheaves of sets on $X$ is denoted $\mathop{\mathit{Sh}}\nolimits (X)$.