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 be a topological space. A presheaf of vector spaces on is a family of vector spaces and a collection of associated linear maps, called restriction maps,
Given such that and one often writes rather than .
Definition 2 (Sheaf). Let be a presheaf on a topological space . We call a sheaf on if for all open sets and collections of open sets such that , satisfies the following properties:
- For such that for all , it is given that . (2.1)
- For all collections such that for all there exists such that for all . (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 , and that (2.2) is a statement of analytic continuation.
Definition 3 (Sheaf of holomorphic functions ). Let be a Riemann surface. The presheaf of holomorphic functions on is made up of complex vector spaces of holomorphic functions. For all open sets , is the vector space of holomorphic functions on . The restrictions are the usual restrictions of functions.
Proposition 4 If is a Riemann surface, then is a sheaf on .
Proof. As 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 , they agree on all of .
For (2.2) take some collection such that for all . For , where such that . When , by definition of the . Therefore, is well-defined. Given any , there exists some neighborhood where is holomorphic. From this follows that is holomorphic, which means . ▢
Definition 5 (Direct limit of algebraic objects). Let be a directed set. Let be a family of objects indexed by and be a homomorphism for all with the following properties:
- is the identity of , and
- for all .
Then the pair is called a direct system over .
The direct limit of the direct system is denoted by and is defined as follows. Its underlying set is the disjoint union of the s modulo a certain equivalence relation :
Here, if and , then iff there is some with such that .
More concretely, using the sheaf of holomorphic functions on a Riemann surface, we see that here, the indices correspond to open sets with meaning , and is the restriction . Two holomorphic functions defined on and , represented by and are considered equivalent iff they are equal restricted to some .
Fix a point and requires that the open sets in consideration are the neighborhoods of it. The direct limit in this case is called the stalk of at , denoted . For each neighborhood of , the canonical morphism associates to a section of over an element of the stalk called the germ of at .
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 be a directed set. Let be a family of objects indexed by and be a homomorphism for all with the following properties:
- is the identity of , and
- for all .
Then the pair is an inverse system of groups and morphisms over , and the morphism are called the transition morphisms of the system.
We define the inverse limit of the inverse system as a particular subgroup of the direct product of the s:
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.