---- > [!definition] Definition. ([[constant sheaf]]) > Let $\mathsf{C}$ be a reasonable [[category]]; for simplicity can assume $\mathsf{C}=\mathsf{Set}$. $S$ is an object of $\mathsf{C}$. Let $X$ be a [[topological space]]. Define a $\mathsf{C}$-valued [[sheaf]] $\underline{S}$ on $X$ by taking ($U\subset X$ open) > $\underline{S}(U):= \{ \text{locally constant maps } U \to S\}$ or equivalently endow $S$ with the [[discrete topology]] and have $\mathcal{F}(U):= \{ \text{continuous maps }U \to S \}.$ This is called the **constant sheaf with values in $S$**. It is the [[sheafification]] of the [[constant presheaf]].[^1] ^definition > [!basicexample] > - For $U \subset X$ [[connected]], [[locally constant implies globally constant on connected space|one has]] $\underline{S}(U)= \{ \text{constant maps } U \to S \}.$ > > - For $X$ [[irreducible topological space|irreducible]] (e.g., $X$ an [[integral scheme|integral scheme]]), one has $\underline{S}(U)=\begin{cases} > (U \subset X) \mapsto S & U \neq \emptyset \\ > \emptyset \mapsto 0 > \end{cases},$ > i.e., $\underline{S}$ with the [[constant presheaf]] $\underline{S}_{\text{pre}}$ for all nonempty opens. > > Indeed, for $X$ [[irreducible topological space|irreducible]] and $U \subset X$ a nonempty open subset, $U$ is also irreducible, hence connected, so any element of $\underline{S}(U)$ is a constant map $U \to S$. There is a [[bijection]] between constant maps $U \to S$ and elements of $S$. ^bea7f6 [^1]: Think about $\mathcal{F}^{+}(U)=\left\{ s:U \to \coprod_{p \in U} \overbrace{ \mathcal{F}_{p} }^{ =S } : \begin{align} &\textcolor{Thistle}{(1) \ s(p) \in \mathcal{F}_{p} =S\text{ for all }p} \\ & \textcolor{Skyblue}{(2) \ \forall p \in U, \exists \text{nbhd } p \in V \subset U \text{ and }} \\ & \quad \quad \textcolor{Skyblue}{ t \in \overbrace{ \mathcal{F}(V) }^{ =S } \text{ s.t. } s(q)=[V,t] \ \forall q \in V} \end{align} \right\}.$ is clearly identified with the set of locally constant maps $U \to S$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```