1. Stalks

Let be a presheaf on . Let be an element of . The stalk of at , written , is defined to be the colimit

A few remarks about this definition. First, in the last post, we defined the limit of a diagram (functor) indexed . In the above definition, we’re implicitly using the subcategory of of open subsets of containing and the functor is just restricted . (Remember, a sheaf is a functor from .)

Second, if we expand the definition of the colimit, we obtain the following definition of the stalk. The stalk is the set of sections , , modulo the relation for if there is an open set such that . That is, the stalk is equivalence classes of sections over open sets containing modulo restriction. Stalks can be thought of as capturing the local (infinitesimal) behavior of sections.

Example 1 If are topological spaces and is the sheaf on of maps to , then the stalk is the set of germs at of maps .

Let be a map of sheaves. induces a map of stalks. can be defined abstractly using the universal property of the colimit. In terms of equivalence classes of sections, is just given by .

Stalks are useful because often properties of morphisms of sheaves (global functions) can be understood by considering the induced maps of stalks (local behavior) at each point. More on this later.

2. Sheafification

Given a presheaf, when can it be made into a sheaf? The answer is: always. Here’s the main result:

Proposition 1 Let be a presheaf. There is a sheaf and a morphism such that is a isomorphism for all and is universal for maps from . In this case, universal means that if is a map to a sheaf , then there is a unique map such that .

That is, every morphism from to a sheaf factors uniquely through . Universality as usual implies that is unique up to unique isomorphism.

I won’t give the sheafification construction now.

Let be the forgetful functor. (Remember, the category of sheaves on is a full subcategory of the category of presheaves on .) Then sheafification is left adjoint to , that is, there a natural isomorphism

This is a standard picture. We can think of as the “free” sheaf generated by . In general, the free functor is left adjoint to the forgetful functor.

What is sheafification good for? Often, it is natural to make constructions of sheaves “open-set-wise,”, that is, for each . For example, if is a morphism of sheaves, we can define a new sheaf by . Unfortunately, this is not in general a sheaf. But it is a presheaf. Sheafification gives us a way to get back to the category of sheaves when constructions lose the sheaf condition.

The next step is to talk about operations on sheaves. However, next time I’m going to introduce Čech cohomology and start talking about characteristic classes.