Presheaf
المؤلف:
Hartshorne, R
المصدر:
Algebraic Geometry. Berlin: Springer-Verlag
الجزء والصفحة:
pp. 60-61
15-5-2021
1905
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.
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