For a topological space, the presheaf of Abelian groups (rings, ...) on is defined such that
1. For every open subset , an Abelian group (ring, ...) , and
2. For every inclusion of open subsets of , a morphism of Abelian groups (rings, ...)
subject to the conditions:
1. If denotes the empty set, then ,
2. is the identity map , and
3. If are three open subsets, then .
In the language of categories, let be the category whose objects are the open subsets of and the only morphisms are the inclusion maps. Thus, is empty if and has just one element if . Then a presheaf is a contravariant functor from the category to the category of Abelian groups ( of rings, ...).
As a terminology, if is a presheaf on , then are called the sections of the presheaf over the open set , sometimes denoted as . The maps are called the restriction maps. If , then the notation is usually used instead of .