Visualising sheaves and presheaves

[adammclean]

I am struggling to get an intuitive grasp of pre-sheaves and sheaves.
It would really help me if I could find some visualisation or pictorial representation of these mathematical ideas.

Of course, in areas of great abstraction these visualisations may fail to capture the full sweep of the ideas involved, but it would be a start at least for me. By merely reading and reading the formalism I am not making much progress.

There are some neat books with good pictorial representations of topological ideas, but with sheaves and cohomology it seems few writers have found pictures they feel might illustrate the concepts.

Can anyone point me to any attempts at depicting these ideas

 

[Hurkyl]

The notion "sheaf on a topological space X" is equivalent to the notion "a space with a local homeomorphism to X" -- however, the space is usually non-Hausdorff and doesn't easily lend itself to being visualized.

There are exceptions, of course. e.g. the sheaf of sections of a covering space corresponds to the covering space itself with the cover. Or the sheaf on C given by

S(U) = the set of all analytic functions f satisfying $z = e^{f(z)}$ on U​

corresponds to the Riemann surface for log z.


Instead of thinking of a space as a set of points, you could instead look at it as a lattice of open sets -- and you don't often need to look at all of them.

A simple example: You want to understand the sheaf of continuous real-valued functions on the circle? Well, if we understand the sheaf of continuous real-valued functions on the line, we can draw the diagram

$$\begin{matrix}& & S^1<br /> \\ & \uprightarrow & & \upleftarrow<br /> \\ R & & & R<br /> \\ & \upleftarrow & & \uprightarrow<br /> \\ & & R^*$$​

where the two copies of R are the open sets corresponding to removing a single point, and R^* is their intersection, which is homeomorphic to the real line minus a point (which, of course, is homeomorphic to two copies of R)

We understand this diagram as showing how the circle is constructed by gluing together copies of the real line. Well, if we pass to the sheaf:

$$\begin{matrix}& & C(S^1)<br /> \\ & \downleftarrow & & \downrightarrow<br /> \\ C(R) & & & C(R)<br /> \\ & \downrightarrow & & \downleftarrow<br /> \\ & & C(R^*)$$​

which encodes some of the relationship of the sheaf C(S¹) to other sheaves that we understand.


The full sweep of sheaf theory is vast -- pushed to its extreme (topos theory), it encompasses both set theory, topology, and generalizations of both.

(P.S. there are supposed to be arrows in those diagrams, upward angled in the first, downward angled in the second, but I don't remember how to draw them)