ABSTRACT
The category of presheaves on a topological space X is a functor category: we turn the topological space into a category C having the open sets in X as objects and a single morphism from U to V if and only if U is contained in V. The category of presheaves of sets (abelian groups, rings) on X is then the same as the category of contravariant functors from C to Set. Because of this example, the category Funct(Cop, Set) is sometimes called the “category of presheaves of sets on C” even for general categories C not arising from a topological space. To define sheaves on a general category C, one needs more structure: a Grothendieck topology on. Categories that are equivalent to Set C are at times called presheaf categories.
