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In the preceding subsection, we have not been explained what ``sheafification''
really means. Here is the definition.
LEMMA 06.12
Let
be a presheaf on a topological space
.
Then there exists a sheaf
and a presheaf morphism
such that the following property holds.
- If there is another sheaf
with a presheaf morphism
then there exists a unique sheaf homomorphism
such that
holds.
Furthermore, such
is unique.
DEFINITION 06.13
The sheaf

(together with

)
is called the sheafification of

.
The proof of Lemma 6.12 is divided in steps.
The first step is to know the uniqueness of such sheafification.
It is most easily done by using universality arguments.
([#!Lang1!#] has a short explanation on this topic.)
Then we divide the sheafification process in two steps.
LEMMA 06.14 (First step of sheafification)
Let
be a presheaf on a topological space
.
Then for each open set
,
we may define a equivalence relation on
by
Then we define
Then
is a presheaf that satisfies the locality axiom
of a sheaf. There is also a presheaf homomorphism from
to
. Furthermore,
is universal among such.
Proofs of the above two lemma are routine work
and are left to the reader.
Finish of the proof of Lemma 6.12:
We put
ARRAY(0xab3bc20)
Next: stalk of a presheaf
Up: Resolutions of singularities.
Previous: example of presheaves and
2014-07-03