§ 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
- 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.
Next, to be able to combine germs together, we need more.
- Sheaf: Adds data to a presheaf to glue functions.
§ A presheaf that is not a sheaf: Bounded functions
Consider the function f(x)≡x. This is bounded on every open interval
I≡(l,r): l≤f(I)≤f(r) But the full function f(x) is unbounded.
§ Holomorphic function with holomorphic square root.
Our old enemy, monodromy shows up here.
Consider the identity function f(z)=z. Let's analyze its square root
on the unit circle. f(eiθ)=eiθ/2. This can only be defined
continuously for half the circle. As we go from θ:0→2π,
our z goes from 0→0, while f(z) goes 0→−1. This
gives us a discontinuity at 0.
§ Formalisms
- Sections of a presheaf F over an open set U:For each open set U⊆X, we have a set F(U), which are generally sets of functions.The elements of F(U) are called as the Sections of F over U.More formally, we have a function F:τ→(τ→R) (τ→R) is the space of functions over τ.
- Restriction Map: For each inclusion U↪V, (U⊆V)we have a restriction map Res(V,U):F(V)→F(U).
- Identity Restriction: The map Res(U,U) is the identity map.
- Restrictions Compose: If we have U⊆V⊆W, we must have Res(W,U)=Res(W,V)∘Res(V,U).
- Germ: A germ of a point p is any section over any open set U containing p.That is, the set of all germs of p is formally Germs(p)≡{F(Up):Up⊆X,p∈U,U open}.We sometimes write the above set as Germs(p)≡{(f,Up):f∈F(Up),Up⊆X,p∈U,U open}.This way, we know both the function f and the open set U over which it is defined.
- Stalk: A stalk at a point p, denoted as Fp,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 (f,U)∼(g,V) iff there exists a W⊆U∩V such thatthe functions f and g agree on W: Res(U,W)(f)=Res(V,W)(g).
- Stalk as Colimit: We can also define the stalk as a colimit. We take theindex category J as a filtered set. Given any two open sets U,V, we have a smalleropen set that is contained in U∩V. This is because both U and V cannotbe non-empty since they share the point p.
- If p∈U and f∈F(U), then the image of f in Fp, as in, the valuethat corresponds to f in the stalk is called as the germ of f at p.This is really confusing! What does this mean?I asked on
math.se
.
§ References
- The rising sea by Ravi Vakil.