Sheaves of holomorphic functions

I can sense vaguely that the sheaf is a central definition in the (superficially) horrendously abstract language of modern mathematics. There really does seem to be quite a distance, between crudely speaking, pre-1950 math and post-1950 math in the mainstream in terms of the level of abstraction typically employed. It is my hope that I will eventually accustom myself to the latter instead of viewing it as a very much alien language. It is difficult though, and  there are in fact definitions which take quite me a while to grasp (by this, I mean be able to visualize it so clearly that feel like I won’t ever forget it), which is expected given how long it has taken historically to condense to certain definitions golden in hindsight. In the hope of a step forward in my goal to understand sheaves, I’ll write up the associated definitions in this post.

Definition 1 (Presheaf). Let $(X, \mathcal{T})$ be a topological space. A presheaf of vector spaces on $X$ is a family $\mathcal{F} = \{\mathcal{F}\}_{U \in \mathcal{T}}$ of vector spaces and a collection of associated linear maps, called restriction maps,

$\rho = \{\rho_V^U : \mathcal{F}(U) \to \mathcal{F}(V) | V,U \in \mathcal{T} \text{ and } V \subset U\}$

such that

$\rho_U^U = \text{id}_{\mathcal{F}(U)} \text{ for all } U \in \mathcal{T}$

$\rho_W^V \circ \rho_V^U = \rho_W^U \text{ for all } U,V,W \in \mathcal{T} \text{ such that } W \subseteq V \subseteq U$.

Given $U,V \in \mathcal{T}$ such that $V \subseteq U$ and $f \in \mathcal{F}(U)$ one often writes $f|_V$ rather than $\rho_V^U(f)$.

Definition 2 (Sheaf). Let $\mathcal{F}$ be a presheaf on a topological space $X$. We call $\mathcal{F}$ a sheaf on $X$ if for all open sets $U \subseteq X$ and collections of open sets $\{U_i \subseteq U\}_{i \in I}$ such that $\cup_{i \in I} U_i = U$, $\mathcal{F}(U)$ satisfies the following properties:

1. For $f, g \in F(U)$ such that $f|_{U_i} = g|_{U_i}$ for all $i \in I$, it is given that $f = g$.    (2.1)
2. For all collections $\{f_i \in F(U_i)\}_{i \in I}$ such that $f_i |_{U_i \cap U_j} = f_j |_{U_i \cap U_j}$ for all $i, j \in I$ there exists $f \in F(U)$ such that $f |_{U_i} = f_i$ for all $i \in I$.    (2.2)

In more concrete terms, it is not difficult to see that (2.1) is a statement of power series about a point with radius of convergence covering $U$, and that (2.2) is a statement of analytic continuation.

Definition 3 (Sheaf of holomorphic functions $\mathcal{O}$). Let $X$ be a Riemann surface. The presheaf $\mathcal{O}$ of holomorphic functions on $X$ is made up of complex vector spaces of holomorphic functions. For all open sets $U \subseteq X$, $\mathcal{O}(U)$ is the vector space of holomorphic functions on $U$. The restrictions are the usual restrictions of functions.

Proposition 4  If $X$ is a Riemann surface, then $\mathcal{O}$ is a sheaf on $X$.

Proof. As $\mathcal{O}$ is a presheaf, it suffices to show properties (2.1) and (2.2)(2.1) follows directly from the definition of restriction of a function. If they agree on every set in the cover of $U$, they agree on all of $U$.

For (2.2) take some collection $\{f_i \in \mathcal{O}(U_i)\}_{i \in I}$ such that $f_i |_{U_i \cap U_j} = f_j |_{U_i \cap U_j}$ for all $i, j \in I$. For $x \in U$, $f(x) = f_i(x)$ where $i \in I$ such that $x \in U$. When $\in U_i \cap U_j$$f_i |_{U_i \cap U_j} = f_j |_{U_i \cap U_j}$ by definition of the $f_i$. Therefore, $f$ is well-defined. Given any $x \in U$, there exists some neighborhood $U_i \in \mathcal{U}$ where $f_i$ is holomorphic. From this follows that $f$ is holomorphic, which means $f \in \mathcal{O}(U)$.     ▢

Definition 5 (Direct limit of algebraic objects). Let $\langle I, \leq \rangle$ be a directed set. Let $\{A_i : i \in I\}$ be a family of objects indexed by $I$ and $f_{ij}: A_j \rightarrow A_j$ be a homomorphism for all $i \leq j$ with the following properties:

1. $f_{ii}$ is the identity of $A_i$, and
2. $f_{ik} = f_{jk} \circ f_{ij}$ for all $i \leq j \leq k$.

Then the pair $\langle A_i, f_{ij} \rangle$ is called a direct system over $I$.

The direct limit of the direct system $\langle A_i, f_{ij} \rangle$ is denoted by $\varinjlim A_i$ and is defined as follows. Its underlying set is the disjoint union of the $A_i$s modulo a certain equivalence relation $\sim$:

$\varinjlim A_i = \bigsqcup_i A_i \bigg / \sim$.

Here, if $x_i \in A_i$ and $x_j \in A_j$, then $x_i \sim x_j$ iff there is some $k \in I$ with $i \leq k, j \leq k$ such that $f_{ik}(x_i) = f_{jk}(x_j)$.

More concretely, using the sheaf of holomorphic functions on a Riemann surface, we see that here, the indices correspond to open sets with $i \leq j$ meaning $U \supset V$, and $f_{ij} : A_i \to A_j$ is the restriction $\rho_V^U : \mathcal{F}(U) \to \mathcal{F}(V)$. Two holomorphic functions defined on $U$ and $V$, represented by $x_i$ and $x_j$ are considered equivalent iff they are equal restricted to some $W \subset V \cap U$.

Fix a point $x \in X$ and requires that the open sets in consideration are the neighborhoods of it. The direct limit in this case is called the stalk of $F$ at $x$, denoted $F_x$. For each neighborhood $U$ of $x$, the canonical morphism $F(U) \to F_x$ associates to a section $s$ of $F$ over $U$ an element $s_x$ of the stalk $F_x$ called the germ of $s$ at $x$.

Dually, there is the inverse limit, which in our concrete context is the more abstract language for an analytic continuation.

Definition 6 (Inverse limit of algebraic objects). Let $\langle I, \leq \rangle$ be a directed set. Let $\{A_i : i \in I\}$ be a family of objects indexed by $I$ and $f_{ij}: A_j \rightarrow A_j$ be a homomorphism for all $i \leq j$ with the following properties:

1. $f_{ii}$ is the identity of $A_i$, and
2. $f_{ik} = f_{jk} \circ f_{ij}$ for all $i \leq j \leq k$.

Then the pair $((A_i)_{i \in I}, (f_{ij})_{i \leq j \in I})$ is an inverse system of groups and morphisms over $I$, and the morphism $f_{ij}$ are called the transition morphisms of the system.

We define the inverse limit of the inverse system $((A_i)_{i \in I}, (f_{ij})_{i \leq j \in I})$ as a particular subgroup of the direct product of the $A_i$s:

$A = \displaystyle\varprojlim_{i \in I} A_i = \left\{\left.\vec{a} \in \prod_{i \in I} A_i\; \right|\;a_i = f_{ij}(a_j) \text{ for all } i \leq j \text{ in } I\right\}$.

What we have essentially are families of holomorphic functions over open sets, and we glue them together via a direct product indexed by open sets under the restriction there must be agreement in values at places where the open sets coincide. This gives us the space of holomorphic functions over the union of the open sets, which is of course a subgroup of the direct product under both addition and multiplication. We have here again the common theme of patching up local pieces to create a global structure.