The main reference: Hartshorne.
A topological space can be seen as a category if we take open sets as its objects and inclusion maps as its morphisms, i.e.
Definition 1: A presheaf is a functor , where is the dual category of a topological space and is the category of Abelian groups. In other words, is a contravariant functor. The elements are called the sections on . Denote that and for .
The morphisms between two presheaf and is just the morphisms between two functors, that is, there exist a family of morphisms such that the following diagram is commutative.
Definition 2: If is a presheaf on , and if is a point on , we define the stalk of at to be the direct limit of the group for all containing , via the restriction maps .
We can directly construct that where is defined as follows: Suppose and , if there exists a such that , then or . is indeed a group since the addition can be defined by .
Here defines a family of morphisms for any . We can then vertify the universal property of direct product that
Now, suppose is a functor. The diagram
is commutative, then the universal property of direct product
gives the existence of the morphism , and the diagram
Definition 3: A sheaf is a presheaf that for any open set , the complex is exact for any open cover of , where
The definition can be rewritten as: For any open cover of any open set ,
If , , then .
If , , then there’s a section such that .
It is not so difficult to vertify that this definition is equivalent to the old one.
The following proposition (which would be false for presheves) illustrates the local nature of a sheaf.
Propostion 1: Suppose is a morphism of sheaves on a topological space . Then is a isomorphism if and only if the induced map on the stalk is an isomorphism for every .
p.63 on Hartshorne.
Definition 4: Let be a morphism of presheaves. We define the presheaf kernel of , presheaf of cokernel of , and presheaf image of to be the presheaves given by , , and respectively.
If is a morphism of sheaves, then the presheaf is a sheaf.
be an open cover of
is local section on
Suppose and , since is a sheaf, .
Suppose , we need to show that there exists a global section such that . Since is a sheaf, it’s nature that there’s such that . The last thing to vetify is . Restrict on , then so vanishes locally. Since is a sheaf, it also vanishes globally, i.e. .
is a sheaf.
However, the presheaves and need not to be sheaves. Actually, the key point in the proof above is that is compatible with the sheaf property of . Then we come to an important notion of a sheaf associated to a presheaf, i.e. sheafification.
Roughly speaking, the sheafification of a presheaf is the “smallest” sheaf with the same stalks as . Because of the “smallest”, sheafification should have the universal property.
Propostion 3: Given a presheaf , there is a sheaf and a morphism make the diagram
commutative for any sheaf . is called the sheaf associated to the preshead or the sheafification of .
We construct the sheaf
as follows. For any open set
be the set of functions
, such that
for each , , and
for each , there is a neighborhood of , and an element , such that , .
The addition on is that , so is indeed a group. If , there’s a nature map (function restriction) such that , so is a presheaf. Let be a open cover of and be local sections. If , we can define a function by setting . If for all , then since it is a function. Thus is a sheaf.
For each , we can associate it a section by , then there’s a morphism . , since and is a morphism.
, because of the construction of
, we can find an open cover
. Firstly define
, and we can use the sheaf condition of
to get a global section
. Then, define
will become the morphism
. Finally, because of the construction of
, for any
, so that
makes the diagram commutative.
Proposition 4: , so if is a sheaf, then .
There's a morphism
, such that
It is injective. If , then , and .
It is surjective.
, there exists an open subset
Definiton 5: If is a morphism of sheaves, we define the kernel(repsectively cokernel, image) of , denoted (repsectively , ), to be the sheaf associated to the presheaf of kernel(respectively, coker, image) of .
Definiton 6: We say that a morphism of sheaves is injective(respectively, surjective) if (respectively, ).
Propositon 5: For any morphism of sheaves , and for each .
We will prove it in the following steps:
Suppose , then , so there exists such that thus and .
Suppose , then , thus .
Suppose and , then .
Suppose , then .
Corollary 1: For any morphism of sheaves , it is injective(respectively, surjective) if and only if is injective(respectively, surjective) for all .
According to Proposition 1 and Proposition 5,
if and only if
if and only if
Corollary 2: A morphism of sheaves is an isomorphism if and only if it is injective and surjective.
A morphism of sheaves
is an isomorphism if and only if
is an isomorphism for all
. As a morphism of groups,
is an isomorphism for all
if and only if it is injective and surjective for all
, and according to Corollay 1, if and only if
is injective and surjective.
Definition 7: Let be a continuous map of topological spoaces. For any sheaf on , we define the direct image sheaf on by for any open set . For any sheaf on , we define the inverse image sheaf on to be the sheaf associated to the presheaf , where is any open set in , and the limit is taken over all open sets containing .