For a topological space a presheaf with values in a category associates to each open set , an object of and to each inclusion a morphism of , , the restriction morphism. It is required that and for any .

A presheaf with values in the category of sets (or abelian groups) is called a presheaf of sets (or abelian groups). If no target category is specified, either the category of sets or abelian groups is most likely understood.

A more categorical way to state it is as follows. For form the category whose objects are open sets of and whose morphisms are the inclusions. Then a presheaf is merely a contravariant functor .