Definition:Presheaf on Topological Space

Definition
Let $X$ be a topological space.

Let $\mathbf C$ be a category.

Definition 1
Let $\tau$ be the set of open subsets of $X$.

A $\mathbf C$-valued presheaf on $X$ is a pair $(\mathcal F, \operatorname{res})$ where:
 * $\mathcal F$ is a mapping on $\tau$ whose image consists of objects of $\mathbf C$.
 * $\operatorname{res}$ is a mapping on $\{(U,V) \in \tau^2 : U\supseteq V\}$ such that for all $U,V,W \in \tau$ with $U\supseteq V\supseteq W$:
 * $\operatorname{res}^{U}_V$ is a morphism from $\mathcal F(U)$ to $\mathcal F(V)$
 * $\operatorname{res}^{U}_U = \operatorname{id}_{\mathcal F(U)}$, the identity morphism on $\mathcal F(U)$
 * $\operatorname{res}^{U}_V \circ \operatorname{res}^{V}_W= \operatorname{res}^{U}_W$, where $\circ$ is the composition in $\mathbf C$

Definition 2
Let $\tau$ be the order category of open subsets of $X$ ordered by inclusion.

A $\mathbf C$-valued presheaf on $X$ 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