Definition:Sheaf on Topological Space/Definition 1

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

Let $\mathbf C$ be a category.

A $\mathbf C$-valued sheaf $\mathcal F$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subseteq S$ and all open covers $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of $U$:
 * $\left({\mathcal F \left({U}\right), \left({\operatorname{res}^U_{U_i} }\right)_{i \mathop \in I} }\right)$

is the limit of the restriction of $\mathcal F$ to $\left\{ {U}\right\} \cup \left\{ {U_i: i \in I}\right\} \cup \left\{ {U_i \cap U_j : \left({i, j}\right) \in I^2}\right\}$

Also see

 * Equivalence of Definitions of Sheaf on Topological Space