Definition:Constant Presheaf

Definition
Let $X$ be a topological space.

Let $S$ be a set.

The constant presheaf with value $S$ on $X$ is the set-valued presheaf
 * $\ds F : \map {\mathbf{Ouv} }{X}^{\mathrm{op} } \to \mathbf{Set}$

from the category of open sets $\map {\mathbf{Ouv}}{X}$ of $X$ to the category of sets $\mathbf{Set}$, defined as follows:


 * For each open subset $U \subset X$, let $\map F U := S$
 * For each inclusion map $i : U \to V$, let $\map F i := \operatorname{id}_S$.

Also see

 * Definition:Constant Sheaf
 * Sheafification of Constant Presheaf is Constant Sheaf
 * Universal Property of Constant Presheaf

Also defined as
Some authors require, that $\map F \emptyset$ is a singleton set. However this is not necessary and a common source of confusion.