Definition:Presheaf 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 presheaf on $T$ is a pair $\left({\mathcal F, \operatorname{res} }\right)$ where:
 * $\mathcal F$ is a mapping on $\tau$ whose image consists of objects of $\mathbf C$


 * $\operatorname{res}$ is a mapping on $\left\{ {\left({U, V}\right) \in \tau^2: U \supseteq V}\right\}$ such that for all $U, V, W \in \tau$ with $U \supseteq V \supseteq W$:
 * $\operatorname{res}_V^U$ is a morphism from $\mathcal F \left({U}\right)$ to $\mathcal F \left({V}\right)$
 * $\operatorname{res}_U^U = \operatorname{id}_{\mathcal F \left({U}\right)}$, the identity morphism on $\mathcal F \left({U}\right)$
 * $\operatorname{res}_V^U \circ \operatorname{res}_W^V = \operatorname{res}_W^U$, where $\circ$ is the composition in $\mathbf C$

Also see

 * Equivalence of Definitions of Presheaf on Topological Space