Definition:Constant Sheaf

Definition

Let $X$ be a topological space.

Let $S$ be a set.

The constant sheaf 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 $\map F U$ be the set of continuous maps $U \to S$ for the discrete topology on $S$.

For each inclusion map $i : U \to V$, let $\map F i : \map F V \to \map F U$ denote restriction of mapping.

Also see

• Sheaf of Continuous Maps is Sheaf

Sources

• 1977: Robin Hartshorne: Algebraic Geometry: $\S \text {II.1}$: Example 1.0.3.