§ 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 $f(x) \equiv x$. This is bounded on every open interval $I \equiv (l, r)$: $l \leq f(I) \leq f(r)$ 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. $f(e^{i \theta}) = e^{i \theta/2}$. This can only be defined continuously for half the circle. As we go from $\theta: 0 \rightarrow 2 \pi$, our $z$ goes from $0 \rightarrow 0$, while $f(z)$ goes $0 \rightarrow -1$. This gives us a discontinuity at $0$.

§ Formalisms

• Sections of a presheaf $F$ over an open set $U$: For each open set $U \subseteq X$, 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: \tau \rightarrow (\tau \rightarrow R)$$(\tau \rightarrow R)$ is the space of functions over $\tau$.
• Restriction Map: For each inclusion $U \hookrightarrow V$, ( $U \subseteq V$) we have a restriction map $Res(V, U): F(V) \rightarrow F(U)$.
• Identity Restriction: The map $Res(U, U)$ is the identity map.
• Restrictions Compose: If we have $U \subseteq V \subseteq W$, we must have $Res(W, U) = Res(W, V) \circ Res(V, U)$.
• 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 $Germs(p) \equiv \{ F(U_p) : U_p \subseteq X, p \in U, U \text{ open} \}$. We sometimes write the above set as $Germs(p) \equiv \{ (f, U_p) : f \in F(U_p), U_p \subseteq X, p \in U, U \text{ open} \}$. This way, we know both the function $f$ and the open set $U$ over which it is defined.
• Stalk: A stalk at a point $p$, denoted as $F_p$, 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 $(f, U) \sim (g, V)$ iff there exists a $W \subseteq U \cap V$ such that the 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 the index category $J$ as a filtered set. Given any two open sets $U, V$, we have a smaller open set that is contained in $U \cap V$. This is because both $U$ and $V$ cannot be non-empty since they share the point $p$.
• If $p \in U$ and $f \in F(U)$, then the image of $f$ in $F_p$, as in, the value that 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.