§ 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)xf(x) \equiv x. This is bounded on every open interval I(l,r)I \equiv (l, r): lf(I)f(r)l \leq f(I) \leq f(r) But the full function f(x)f(x) is unbounded.

§ Holomorphic function with holomorphic square root.

Our old enemy, monodromy shows up here. Consider the identity function f(z)=zf(z) = z. Let's analyze its square root on the unit circle. f(eiθ)=eiθ/2f(e^{i \theta}) = e^{i \theta/2}. This can only be defined continuously for half the circle. As we go from θ:02π\theta: 0 \rightarrow 2 \pi, our zz goes from 000 \rightarrow 0, while f(z)f(z) goes 010 \rightarrow -1. This gives us a discontinuity at 00.

§ Formalisms

  • Sections of a presheaf FF over an open set UU: For each open set UXU \subseteq X, we have a set F(U)F(U), which are generally sets of functions. The elements of F(U)F(U) are called as the Sections of FF over UU. More formally, we have a function F:τ(τR)F: \tau \rightarrow (\tau \rightarrow R)(τR)(\tau \rightarrow R) is the space of functions over τ\tau.
  • Restriction Map: For each inclusion UVU \hookrightarrow V, ( UVU \subseteq V) we have a restriction map Res(V,U):F(V)F(U)Res(V, U): F(V) \rightarrow F(U).
  • Identity Restriction: The map Res(U,U)Res(U, U) is the identity map.
  • Restrictions Compose: If we have UVWU \subseteq V \subseteq W, we must have Res(W,U)=Res(W,V)Res(V,U)Res(W, U) = Res(W, V) \circ Res(V, U).
  • Germ: A germ of a point pp is any section over any open set UU containing pp. That is, the set of all germs of pp is formally Germs(p){F(Up):UpX,pU,U open}Germs(p) \equiv \{ F(U_p) : U_p \subseteq X, p \in U, U \text{ open} \}. We sometimes write the above set as Germs(p){(f,Up):fF(Up),UpX,pU,U open}Germs(p) \equiv \{ (f, U_p) : f \in F(U_p), U_p \subseteq X, p \in U, U \text{ open} \}. This way, we know both the function ff and the open set UU over which it is defined.
  • Stalk: A stalk at a point pp, denoted as FpF_p, 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 (f,U)(g,V)(f, U) \sim (g, V) iff there exists a WUVW \subseteq U \cap V such that the functions ff and gg agree on WW: Res(U,W)(f)=Res(V,W)(g)Res(U, W)(f) = Res(V, W)(g).
  • Stalk as Colimit: We can also define the stalk as a colimit. We take the index category JJ as a filtered set. Given any two open sets U,VU, V, we have a smaller open set that is contained in UVU \cap V. This is because both UU and VV cannot be non-empty since they share the point pp.
  • If pUp \in U and fF(U)f \in F(U), then the image of ff in FpF_p, as in, the value that corresponds to ff in the stalk is called as the germ of ff at pp. This is really confusing! What does this mean? I asked on math.se.

§ References

  • The rising sea by Ravi Vakil.