Definition:Restriction of Presheaf to Open Set

Definition
Let $C$ be a category.

Let $X$ be a topological space.

Let $\mathcal F$ be a $C$-valued presheaf on $X$.

Let $U \subset X$ be open.

The restriction of $\mathcal F$ to $U$ is the restriction of the contravariant functor $\mathcal F$ to the subcategory $U$.

Also see

 * Definition:Restriction of Presheaf to Open Set Functor
 * Restriction of Sheaf to Open Set is Sheaf
 * Definition:Section of Presheaf Functor

Generalizations

 * Definition:Inverse Image Presheaf