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 
.
