§ Germs, Stalks, Sheaves of differentiable functions
I know some differential geometry, so I'll be casting sheaves in terms
of tangent spaces for my own benefit
Next, to be able to combine germs together, we need more.
- Presheaf: Data about restricting functions.
- Germ: Equivalence class of functions in the neighbourhood at a point, which become equivalent on restriction. Example: equivalence classes of curves with the same directional derivative.
- Stalk: An algebraic object worth of germs at a point.
- Sheaf: Adds data to a presheaf to glue functions.
§ A presheaf that is not a sheaf: Bounded functions
Consider the function . This is bounded on every open interval
: But the full function is unbounded.
§ Holomorphic function with holomorphic square root.
Our old enemy, monodromy shows up here.
Consider the identity function . Let's analyze its square root
on the unit circle. . This can only be defined
continuously for half the circle. As we go from ,
our goes from , while goes . This
gives us a discontinuity at .
- Sections of a presheaf over an open set : For each open set , we have a set , which are generally sets of functions. The elements of are called as the Sections of over . More formally, we have a function is the space of functions over .
- Restriction Map: For each inclusion , ( ) we have a restriction map .
- Identity Restriction: The map is the identity map.
- Restrictions Compose: If we have , we must have .
- Germ: A germ of a point is any section over any open set containing . That is, the set of all germs of is formally . We sometimes write the above set as . This way, we know both the function and the open set over which it is defined.
- Stalk: A stalk at a point , denoted as , consists of equivalence classes of all germs at a point, where two germs are equivalent if the germs become equal over a small enough set. We state that iff there exists a such that the functions and agree on : .
- Stalk as Colimit: We can also define the stalk as a colimit. We take the index category as a filtered set. Given any two open sets , we have a smaller open set that is contained in . This is because both and cannot be non-empty since they share the point .
- If and , then the image of in , as in, the value that corresponds to in the stalk is called as the germ of at . This is really confusing! What does this mean? I asked on
- The rising sea by Ravi Vakil.