Pre-sheaf

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

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 $F$ from the 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 on $X$ (cf. Sheaf theory).

Comment

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