Definition:Sheaf on Topological Space/Definition 1

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

Let $\mathbf C$ be a category.

A $\mathbf C$-valued sheaf $\FF$ on $T$ is a $\mathbf C$-valued presheaf such that for all open $U \subseteq S$ and all open covers $\sequence {U_i} _{i \mathop \in I}$ of $U$:
 * $\struct {\map \FF U, \paren {\operatorname {res}_{U_i}^U}_{i \mathop \in I} }$

is the limit of the restriction of $\FF$ to the full subcategory of the category of open sets of $T$ with objects $\set U \cup \set {U_i: i \in I} \cup \set {U_i \cap U_j: \tuple {i, j} \in I^2}$.

Also see

 * Equivalence of Definitions of Sheaf on Topological Space