\begin{definition}[Definition:Sheaf on Topological Space/Definition 1]
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 $\set U \cup \set {U_i: i \in I} \cup \set {U_i \cap U_j: \tuple {i, j} \in I^2}$.
\end{definition}