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 .