Definition:Sheaf on Topological Space/Definition 3

Definition
Let $T = \struct {S, \tau}$ be a topological space.

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

A $\mathbf C$-valued sheaf $\FF$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subset S$ and all open covers $\family {U_i}_{i \mathop \in I}$ of $U$ the sequence:


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

0 \ar[r] & \map F U \ar[r]^r & \prod_{i \mathop \in I} \map \FF {U_i} \ar[r]^{\!\!\!\!\!\!\!\!\! r_1-r_2} & \ds \prod_{\tuple {i, j} \mathop \in I^2} \map \FF {U_i \cap U_j} }\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