Definition:Sheaf on Topological Space/Definition 3

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

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:


 * $\begin{xy}\xymatrix@L+2mu@+1em{

0 \ar[r] & F \left({U}\right) \ar[r]^r & \prod_{i \mathop \in I} \mathcal F \left({U_i}\right) \ar[r]^{\!\!\!\!\!\!\!\!\! r_1-r_2} & \prod_{\left({i ,j}\right) \mathop \in I^2} \mathcal F \left({U_i \cap U_j}\right) }\end{xy}$ is exact.

$r$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U}_{U_i} : \map \FF U \to \map \FF {U_i}$.

$r_1$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U_i}_{U_i \cap U_j} : \map \FF {U_i} \to \map \FF {U_i \cap U_j}$.

$r_2$ is induced by the universal property of the product by the restriction maps $\operatorname{res}^{U_j}_{U_i \cap U_j} : \map \FF {U_j} \to \map \FF {U_i \cap U_j}$.

Also see

 * Equivalence of Definitions of Sheaf on Topological Space