Presheaf

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 .

REFERENCES:

Hartshorne, R. Algebraic Geometry. Berlin: Springer-Verlag, pp. 60-61, 1977.