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A presheaf on a topological space with values in a category (usually with values in Grp, CommRing, or Set), can be associated to a sheaf on in a natural way. In fact, this association is left adjoint to the inclusion functor from the category of sheaves on to the category of presheaves on .
The standard construction of is fairly straightforward: consider the topology on the disjoint union of stalks generated by the collection of sets
where is open, and is the image of in . Let be the natural projection. For every define to be the set of continuous sections of , i.e. . It follows that is a sheaf and there is a natural morphism , such that is a sheaf if, and only if, is an isomorphism, (this is a fairly easy exercise).
This shows that every sheaf arises as the sheaf of sections of some continuous map onto . This also justifies the terminology: an element is called a section. At first glance, it is hard to see how this sheaf relates to . Indeed, the canonical map fails to be either surjective or injective, in general. This says that we cannot “complete” by simply adding more elements, or by glueing together elements; we must do both. Thus, we can can’t view as a subpresheaf of or as a cover. However, there is another natural way to think of .
Let be a sheaf on with values in some category , as above. One of the advantages of having a sheaf on is that a section is uniquely determined by the collection of elements . This can fail for presheaves. Thus, we need to glue together elements such that for all . This is evident in the commutative diagram:
This says that and have the same image in for all . Thus, because is a sheaf, .
Aside from glueing sections together that “should” be equal, we also need to add more elements to our presheaf . Suppose we have an open cover of an open subset : . If we can find a collection , such that , then we should be able to find a “global” section on such that . Because our presheaf may be too “small” and may not contain this element, we need to ensure that our sheafification does. The set is compatible in the obvious sense, so we can glue them together to form a section . This map is continuous because . is open. Indeed, if for some , then there is an open subsets such that and in , so . It is also clear that .
To sum up our construction, we note the following: to construct we find the most “efficient” sheaf such that
- sections that have the same germ at every point are equal
- compatible sets of sections have a common extension.
This gives the following alternative description of :
where , if for all and , . Note that condition and the assumption that this extension is unique is just the sheaf axiom and, hence, it implies .
We claim that this definition of is equivalent to the definition given above. Indeed, we can view each collection as a section in the natural way: if . This is independent of the representative of by . Note that is continuous, as we showed earlier. Further, any section arises in this way. Indeed, let , then for a neighborhood of and an element . Thus, because is continuous there exists . Consider the collection . Because for all , it follows that corresponds to . This shows that the correspondence is surjective. We claim that this correspondence is also injective. Indeed, if and correspond to the same section , then for all . Thus, Finally we note that this correspondence is actually a morphism of presheaves: Restricting a collection to a subset , yields the element and this corresponds to the restricted section .
This new definition has some utility. It helps to illuminate the following claims:
Claim 1 Let be the natural map: . Then is a sheaf if, and only if, is an isomorphism.
Proof: If is a sheaf then each collection can be written uniquely as for some . Conversely, suppose is an isomorphism, let be an open subset, let , and let be a collection such that . Then for all , so . Thus, there exists a unique element such that . Therefore, for each , for all . In particular, for each fixed , , whence, , as desired.
Now, suppose that is a sheaf on . Let be a morphism of presheaves.
Claim 2 There exists a unique map , such that .
Proof: Let be an open subset and let . Observe that the commutative diagrams
ensure that for all . Thus, and so by the sheaf axiom for , there is a unique extension . For convenience, we can view as , with . Now, if is an open set, then
Thus, is a morphism . From the definition of , it is clear that and is unique.