View source for Pre-sheaf

{{MSC|14}}

A ''pre-sheaf on a topological space $X$ with values in a category $\def\cK{ {\mathcal K}}\cK$'' (e.g., the category of sets, groups, modules, rings, etc.)
is a contravariant
[[Functor|functor]] $F$ from the
[[Category|category]] of open sets of $X$ and their natural inclusion mappings into $\cK$. Depending on $\cK$, the functor $F$ is called a pre-sheaf of sets, groups, modules, rings, etc. The morphisms $F(U)\to F(V)$ corresponding to the inclusions $V\subseteq U$ are called restriction homomorphisms.

Every pre-sheaf generates a
[[Sheaf|sheaf]] on $X$ (cf.
[[Sheaf theory|Sheaf theory]]).



====Comment====
More generally, if $\def\cC{ {\mathcal C}}\cC$ is any
[[Small category|small category]], the term  "pre-sheaf on $\cC$"  is used to denote a contravariant (usually set-valued) functor defined on $\cC$ (cf.
[[Site|Site]]).