Definition:Constant Presheaf

From ProofWiki
Jump to navigation Jump to search


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

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.