## Sheaves

Our goal in this section is to introduce the machinery necessary to be able to give local definitions of morphisms in a useful way.

Definition Let X be a topological space. Form a category TOP(X) whose objects are the open subsets of X and whose morphisms are the inclusion maps U V.

Let AB denote the category of abelian groups.

Definition A presheaf on a topological space X is a (contravariant) functor

TOP(X)op AB.

We can give a more concrete description of the data that must be specified in order to define a presheaf F on a topological space X. First, for each open subset U X, you must give an abelian group F(U). Next, whenever you have an inclusion U V of open subsets of X, you must give a group homomorphism

ρVU : F(V) F(U).

We refer to the map ρVU as a restriction map. The restriction maps must fit together to define a functor. In particular, the composite of two restriction maps must be a restriction map, and the restriction associated to the identity map on an open set must be the identity homomorphism. As a final piece of notation, we usually refer to an element s F(U) as a section of F over U. We'll try to explain this terminology later.

Remark Presheaves need not be functors from the topological category of X to abelian groups; we often consider presheaves with values in different categories, such as the category of sets, or the category of commutative rings.

Example
1. Fix a topological space Y. For any other topological space X, we can define a presheaf of sets by taking C(U), for all open U X, to be the set of continuous functions from U to Y. Here the restriction maps are the usual restrictions of a function from one set to a smaller set contained inside it.
2. When Y=R or Y=C with the usual topologies, then the presheaf defined in the previous example takes its values in the category of commutative rings. (Here additiona and multiplication of funtions is performed pointwise.)
3. Fix a differentiable manifold Y. For any other differentiable manifold X, we can define a presheaf by taking D(U) to be the set of differentiable functions from U to Y. Since R can be viewed as a differentiable manifold, we can also get a presheaf of rings .
4. Give R the usual topology, and let X be any topological space. Define a presheaf by taking CON(U) to be the (ring of) constant functions on U with values in R.
5. Similarly, one can take LC(U) to be the (ring of) locally constant real-valued functions on U.
6. Notice that for each open subset U of a topological space X, the set of constant functions on U is a (normal) subgroup of the (abelian) group of all locally constant functions. So, as our final example for the moment, we can define G(U) to be the abelian group of locally constant real-valued functions on U modulo the constant functions. The restriction maps are induced from the usual restrictions of locally constant functions, and are well-defined and functorial because of the universal properties of quotient group homomorphisms.

Remark Recall that one of our principal goals was to introduce a mechanism that would allow us to define functions locally and patch them together. Presheaves aren't quite up to this task. We can see this by looking more carefully at some of the examples we have just introduced. First, let's look at the presheaf of constant real-valued functions. Suppose U can be written as the disjoint union of two open subsets V and W. Look at the section s CON(V) that takes on the constant value 0, and at the section t CON(W) that takes on the constant value 1. Because the intersection V W is empty, the sections s and t restrict to the same (trivial) function on the intersection. However, there is no way to patch these two sections together to define a compatible constant function on the entire space U.

The presheaf G of locally constant functions modulo constant functions illustrates another potential problem. Take any open set U X and any section s G(U). By choosing U appropriately, we can choose a non-constant section s. However, since this section is locally constant, we can cover U by open subsets Vi such that all the restrictions ρUVi(s) are consant, and hence are given by the zero element of G(Vi). That is, all (nontrivial) sections of G are locally indistinguishable (from each other or from the trivial section).

Definition sheaf

Examples revisited.

Definition stalk; germ

Example of regular function on an algebraic variety over an algebraically closed field. Example of analytic functions on a complex manifold. Point being that "germs" are like convergent power series.

Proposition you can check isomorphism on the stalks.

Proposition existence of associated sheaves

Note that this says that sections really are sections.

Definition kernel, cokernel, image, quotient, exactness

Proposition you can check exactness on the stalks.

Remark you have to be careful with surjectivity. That's good, because it explains where cohomology comes from.

Proposition It's enough to define a sheaf by giving a (pre)sheaf on a basis for the open sets.