Definition:Sheaf on Topological Space/Definition 2

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


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

r : F \left({U}\right) \ar[r] & \operatorname{eq} \Big(\prod_{i \mathop \in I} \mathcal F \left({U_i}\right) \ar@<-.5ex>[r]_{r_2} \ar@<.5ex>[r]^{r_1} & \prod_{\left({i ,j}\right) \mathop \in I^2} \mathcal F \left({U_i \cap U_j}\right)\Big) }\end{xy}$ is an isomorphism.

$r_1$ 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 $\operatorname{res}^{U_j}_{U_i \cap U_j} : \map \FF {U_j} \to \map \FF {U_i \cap U_j}$.

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

Also see

 * Equivalence of Definitions of Sheaf on Topological Space