Sheaves of Functions on Manifolds
Now that we’ve talked a bunch about presheaves and sheaves in general, let’s talk about some particular sheaves of use in differential topology. Given a smooth manifold — for whatever we choose smooth to mean — we can define sheaves of real algebras of real-valued functions for every less-stringent definition of smoothness.
In the first case of a bare topological manifold , we have no real sense of differentiability at all, and so it only makes sense to talk about continuous real-valued functions . Given an open set we let be the -algebra of real-valued functions that are defined and continuous on .
Next, if is a manifold, then it not only makes sense to talk about continuous real-valued functions — we can define just as above — but we can also talk about differentiable real-valued functions. Given an open set , we let be the -algebra of continuously-differentiable real valued functions .
As we increase the smoothness of , we can consider smoother and smoother functions. If is a manifold, we can define — the sheaf of -times continuously-differentiable functions. Given an open set , we let be the -algebra of real-valued functions on with continuous derivatives.
Continuing up the latter, if is a manifold, then we can define all of the above sheaves, along with the sheaf of infinitely-differentiable functions. And if is analytic, we can also define the sheaf of analytic functions.
In each case, I’m not going to bother going through the proof that we actually do get sheaves. The core idea is that continuity, differentiability, and analyticity are notions defined locally, point-by-point. Thus if we restrict the domain of such a function we get another function of the same kind, and pasting together functions that agree on their overlaps preserves smoothness. This doesn’t hold, however, for global notions like boundedness — it’s easy to define a collection of functions on an open cover of , each of which is bounded, which define an unbounded function when pasted together.
For each class of manifolds, the sheaf of the smoothest functions we can define has a special place. If is in class — where can be , any finite whole number, , or — then the sheaf is often just written , and is called the “structure sheaf” of . It turns out that most, if not all, of the geometrical properties of are actually bound up within its structure sheaf, and so this is a very important object of study indeed.