Definition:Presheaf on Topological Space/Definition 1
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Definition
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:
- $\operatorname{res}$ 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 $\mathbf C$