Examples of Sheaves

Since the posts of sheaves have been more than a bit confusing, I'm going to take
the time to go through a couple of examples of real sheaves that are used in
algebraic topology and related fields. Todays example will be the most canonical one:
a sheaf of continuous functions over a topological space. This can be done for *any* topological space, because a topological space *must* be continuous and gluable with
other topological spaces.

Let's quickly recall the definition of sheaves. A sheaf *F* is a mapping from open sets in
a topological space **T** to objects in a category **C** that has a set of essential properties:

1. Every open set S in **T** is mapped to an object *F*(S) in **C**.
2. For all pairs of open sets in **T**, there is a *restriction morphism* that provides
a mapping between objects that respects the intersection property of the open sets.
3. The category **C** has a terminal object t, and *F*(∅)=t.
4. For all sets U={u_i} of open sets with compatible section in **T** there is exactly one unique section on which the restriction morphisms agree; more formally, there is exactly one section s∈F(U) such that ∀ i, ρ_ui,U(S)=s_i.

You can't show examples of sheaves without starting with the most basic one: sheaves of continuous functions; that is, a sheaf mapping from topological functions to the category
**Top** of sets with continuous functions as morphisms. We can take any topological space **T**, and define a sheaf of continuous functions on it.

First we'll define the presheaf. Let the presheaf *F* for **T** by doing the following:

* For each open set *O* in **T**, Let *F*(*O*) be the set of *continuous* functions f : *O* → ℜ.
* Given two open sets *O* and *P* such that *P* ⊂ *O*, let the restriction morphism
ρ_*P,O* take each function *f* and map it to the function generated by
restricting the domain of *f* to the set *P*.

To show that *F* is not just a presheaf but a full sheaf, we need to show that the
normalization and gluing axioms hold. Normalization, as usual, is pretty trivial. There's
only one function f : ∅ → ℜ: the empty function. Bingo, normalization.

The gluing axiom is harder. And what it comes down to is going to *look* circular. What we *want* is for the functions in the sheaf to behave in a particular way. So we're going to
*define* the value of the functions included in the sheaf so that they work properly. As long as the functions for an open set *u* *exist* in the set of functions from *u* to ℜ, we
can define the sheaf so that it only includes the functions that have the properties that we want. We'll select exactly the functions that give us a sheaf of continuous functions
over the topology.

So. We start by looking at a set of open sets: U = { u_i }.

For any function f_i on an open set u_i, there is a function
f^\* ∈ F(U) such that f_i = f^\* restricted to
u_i. If we restrict the functions in the sheaf to all be restrictions of continuous
functions over **T**, then each of the *sections* over each open set *u_i are
functions over **T** restricted to *u_i*; and two sections are *compatible*
if they are restrictions of the *same* function over **T**. Then, basically by definition,
the gluing axiom will hold - the sections will agree on overlaps, because they've been constructed that way.

So what does this tell us? Basically that gluing topological spaces together will
always result in a topological space - you can't build a structure that won't
be a valid topological space by gluing topological spaces together.

But it says more than that. What it really says is that the things you can do to
functions, you can do to topologies. So the fact that there is this sheaf of functions that can be constructed over any topological space means that you can do *algebra* on topological spaces. Since you can create continuous functions from a space **T** to ℜ by
adding together two continuous functions from **T** to ℜ, and you can create the additive inverse of a function from **T** to ℜ, that means that you've actually got a sheaf of groups. And do the same trick with multiplication, and you can see that this is also a sheaf of rings!