Pre-sheaf

on a topological space with values in a category (e.g. the category of sets, groups, modules, rings, etc.)

A contravariant functor from the category of open sets of and their natural inclusion mappings into . Depending on , the functor is called a pre-sheaf of sets, groups, modules, rings, etc. The morphisms corresponding to the inclusions are called restriction homomorphisms.

Every pre-sheaf generates a sheaf on (cf. Sheaf theory).

Comments

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