Definition:Presheaf on Topological Space

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $\mathbf C$ be a category.

Definition 1

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 $\mathbf C$

Definition 2

Let $\tau$ be the category of open sets of $T$.

A $\mathbf C$-valued presheaf on $T$ is a contravariant functor $\tau \to \mathbf C$.

Also see

• Equivalence of Definitions of Presheaf on Topological Space
• Definition:Category of Presheaves on Topological Space
• Definition:Sheaf on Topological Space