\begin{definition}[Definition:Presheaf on Topological Space/Definition 1]
Let $T = \struct {S, \tau}$ be a topological space.
Let $\mathbf C$ be a category.
A $\mathbf C$-valued '''presheaf''' on $T$ is a pair $\struct {\FF, \operatorname{res} }$ where:
:$\FF$ is a mapping on $\tau$ whose image consists of objects of $\mathbf C$
:$\operatorname{res}$ is a mapping on $\set {\tuple {U, V} \in \tau^2: U \supseteq V}$ such that for all $U, V, W \in \tau$ with $U \supseteq V \supseteq W$:
::$\operatorname{res}_V^U$ is a morphism from $\map \FF U$ to $\map \FF V$
::$\operatorname{res}_U^U = \operatorname{id}_{\map \FF U}$, the identity morphism on $\map \FF U$
::$\operatorname{res}_V^U \circ \operatorname{res}_W^V = \operatorname{res}_W^U$, where $\circ$ is the composition in $\mathbf C$
\end{definition}