§ 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 areequivalent if the germs become equal over a small enough set.We state that iff there exists a such thatthe functions and agree on : .
- Stalk as Colimit: We can also define the stalk as a colimit. We take theindex category as a filtered set. Given any two open sets , we have a smalleropen set that is contained in . This is because both and cannotbe non-empty since they share the point .
- If and , then the image of in , as in, the valuethat 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.