Definition:Sheaf on Topological Space/Definition 3

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $\mathbf C$ be a category.

Let $\mathbf C$ be a complete abelian category.

A $\mathbf C$-valued sheaf $\mathcal F$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subset S$ and all open covers $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of $U$ the sequence:
 * $0 \to \mathcal F \left({U}\right) \to \displaystyle \prod_{i \mathop \in I} \mathcal F \left({U_i}\right) \to \prod_{\left({i ,j}\right) \mathop \in I^2} \mathcal F \left({U_i \cap U_j}\right)$

is exact.

Also see

 * Equivalence of Definitions of Sheaf on Topological Space