Suppose we have a presheaf of abelian groups over and pick a point . On the collection of all abelian groups defined over some neighbourhood of (disjoint union) we put an equivalence relation which identifies and precisely if there is some open in the intersection where and coincide. (or more precisely, their restrictions coincide). The set of equivalence classes turns out to be an Abelian group called the stalk of the presheaf at .
So, with more fluff introduced, the stalk is all the elements in the presheaf that are defined above any neighbourhood of the point, and counted as the same if they seem to be.
For an open set and a point there is a canonical group morphism which sends an element to its equivalence class. This image is the germ of at .
Let be the presheaf of holomorphic function on . Then is precisely the set of all convergent power series of the form for complex . The germ of some defined on a neighbourhood of is precisely the Taylor series around that point.
Take the constant presheaf where over each open set, the additive group of all integers hovers. Then we have for the construction of the stalk a disjoint union of copies of indiced by all the neighbourhoods of the point. Since all the neighbourhoods contain they all have a non-empty intersection, on which elements agree if they have the same value. So for any point . The germ of at is .
We define exact sequences by looking at the induced maps of the stalks. A sequence of sheave morphisms is exact if for every point on the underlying space, the corresponding induced stalk map sequences are exact.
We can construct a morphism of sheaves of abelian groups from to by sending to the function in . For some point , this induces a stalk map that takes in and sends it to the function defined by the convergent power series . Regardless of , this map is obviously an injection, and so this map is a monomorphism.
Space étalé and sheafifications
Given a presheaf , we construct the set as the disjoint union of all stalks . There’s an natural projection down to the underlying topological space. We can introduce a topology on by constructing, for each open set and each element , the set and take these sets to be a basis of our topology. So an open set is given from the by a sequence of (possibly infinite) unions and finite intersections.
This turns out to be a covering of , i.e. each has an open neighbourhood which is homeomorphic to its image under the projection . We call the topological space with this topology the space étalé of the presheaf .
Using the space étalé, then, we can construct a canonical sheaf. A continuous section of a covering space over a subset is a continuous map such that .
is a covering space of the circle (viewed as the interval [0,1] with 0 identified with 1) with the projection . A continuous section of the upper open halfcircle is a map . Indeed, takes some point to and then to
Now, let denote the set of all continuous sections of over . This ends up in the same category that goes to. From this, we then define a sheaf by setting , and letting restriction be the usual restriction of maps. This gives us a functor from presheaves to sheaves called sheafification.
We already saw that assigning the group to each open subset of a manifold and with all restrictions being identity morphisms is not a sheaf. What happens if we sheafify? First, we need to construct our space étalé. This is the disjoint union of all stalks. A stalk over a point is the quotient of the disjoint union of all for with the equivalence relation that identifies if there is some such that . Now, all neighbourhoods of intersect in some open subset, and since all restrictions are identities, we identify with for all pairs of open neighbourhoods ,; so in end-effect all stalks are isomorphic, as groups, to .
The space étalé is thus the disjoint union for all points of copies of ; i.e. the set of ordered pairs on the form for and . Finally, our sheafified sheaf assigns to each open subset the group of sections , thus a point there is a function such that for some . Since this map has to be continuous, the map is constant on neighbourhoods. And thus we recover the sheaf of locally constant maps for the constant sheaf; just as exhibited earlier.
Kernels and quotients
Let be a map of sheaves. For any open , we define . The sheaf formed by these groups together with the induced morphisms from is called the kernel of the sheaf map.
The quotient of a sheaf map is formed almost the same way – pointwise components are formed as expected, but the presheaf thus formed need not be a sheaf. So we sheafify it.