Definition:Constant Presheaf
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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 $\map F \O$ is a singleton set. However this is not necessary and a common source of confusion.