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_jf_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_is 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_is:

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.

Path lifting lemma and fundamental group of circle

I’ve been reading some algebraic topology lately. It is horrendously abstract, at least for me at my current stage. Nonetheless, I’ve managed to make a little progress. On that, I’ll say that the path lifting lemma, a beautiful fundamental result in the field, makes more sense to me now at the formal level, where as perceived by me right now, the difficulty lies largely in the formalisms.

Path lifting lemma:    Let p : \tilde{X} \to X be a covering projection and \gamma : [0,1] \to X be a path such that for some x_0 \in X and \tilde{x} \in \tilde{X},

\gamma(0) = x_0 = p(\tilde{x_0}). \ \ \ \ (1)

Then there exists a unique path \tilde{\gamma} : [0,1] \to \tilde{X} such that

p \circ \tilde{\gamma} = \gamma, \qquad \tilde{y}(0) = \tilde{x_0}. \ \ \ \ (2)

How to prove this at a high level? First, we use the Lebesgue number lemma on an open cover of X by evenly covered open sets to partition [0,1] into intervals of length 1/n < \eta, with \eta the Lebesgue number, to induce n pieces of the path in X which all lie in some open set of the cover. Because every open set is evenly covered, we for each piece have a uniquely determined continuous map (by the homeomorphism of the covering map plus boundary condition). Glue them together to get the lifted path, via the gluing lemma.

Let \mathcal{O} be our cover of X by evenly covered open sets. Let \eta > 0 be a Lebesgue number for \gamma^{-1}(\mathcal{O}), with n such that 1/n < \eta.

Let \gamma_j be \gamma restricted to [\frac{j}{n}, \frac{j+1}{n}]. At j = 0, we have that p^{-1}(\gamma_0([0,\frac{1}{n}])) consists of disjoint sets each of which is homeomorphic to \gamma_0([0, \frac{1}{n}]), and we pick the one that contains \tilde{x_0}, letting q_0 denote the associated map for that, to \tilde{X}, so that p \circ (q_0 \circ \gamma_0) = \gamma_0, with \tilde{\gamma_0} = q_0 \circ \gamma_0.

We continue like this for j up to n-1, using the value imposed on the boundary, which we have by induction to determine the homeomorphism associated with the covering projection that keeps the path continuous, which we call q_j. With this, we have

\tilde{\gamma_j} = q_j \circ \gamma_j.

A continuous path \tilde{\gamma} is obtained by applying to the gluing lemma to these. That

p \circ \tilde{\gamma} = \gamma

is satisfied because it is satisfied on sets the union of which is the entire domain, namely \{[\frac{j}{n}, \frac{j+1}{n}] : j = 0,1,\ldots,n-1\}.

A canonical example of path lifting is that of lifting a path on the unit circle to a path on the real line. To every point on the unit circle is associated its preimage under the map t \mapsto (\cos t, \sin t). It is not hard to verify that this is in fact a covering space. By the path lifting lemma, there is some unique path on the real line that projects to our path on the circle that ends at some integer multiple of 2\pi, call it 2\pi n, and that path is homotopic to the direct path from 0 to 2\pi n via the linear homotopy. Application of the projection onto that homotopy yields that our path on the circle, which we call f, is homotopic to the path where one winds around the circle n times counterclockwise, which we call \omega_n.

Homotopy between f and \omega_n is unique. If on the other hand, \omega_n were homotopic to \omega_m for m \neq n, they we could lift the homotopy onto the real line, thereby yielding a contradiction as there the endpoints would not be the same.

This requires a homotopy lifting lemma. The proof of that is similar to that of path lifting, but it is more complicated, since there is an additional homotopy parameter, by convention, within [0,1], alongside the path parameter. Again, we use the Lebesgue number lemma, but this time on grid [0,1] \times [0,1], and again for each grid component there is a unique way to select the local homeomorphism such that there is agreement with its neighboring components, with the parameter space in common here an edge common to two adjacent grid components.

With that every path on the circle is uniquely equivalent by homotopy to some unique \omega_n, we have that its fundamental group is \mathbb{Z}, since clearly, \omega_m * \omega_n = \omega_{m+n}, where here, * is the path concatenation operation.