A Universe of Sorts

§ 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.