What Are Sheaves?

In 1940, the French mathematician and artillery officer Jean Leray was taken prisoner by the Germans. He told his captors that he was a topologist, fearful that if they discovered his true area of expertise, hydrodynamics, they would force him to aid the German war effort. For the nearly five years of his imprisonment, Leray kept up this subterfuge by carrying out research in topology, a branch of mathematics that studies deformable shapes. He wound up creating one of the most revolutionary ideas in modern mathematics: the notion of a “sheaf.”

After Alexander Grothendieck brought Leray’s notion to prominence in the 1950s and ’60s, sheaves took on a “starring role” in mathematics, said David Ben-Zvi of the University of Texas, Austin, becoming “one of the most basic tools in modern algebraic geometry.”

As one introductory explanation puts it, sheaves can be thought of as developments built on top of other mathematical objects. “Think about it like the mathematical object is a plot of land and a sheaf is like a garden on top of it,” Mark Agrios wrote.

Sheaves got their name because they involve attaching “stalks” to an underlying object. Leray named them “faisceaux” (French for “sheaves”) because this arrangement reminded him of bundles of harvested wheat. Just as gardens can be grown on different kinds of land, sheaves can be built on top of numerous different types of mathematical objects, and therefore can take many different forms.

Even the simplest sheaves are fairly complicated mathematical entities. To better understand them, we can construct one. Here’s how to make a simple sheaf out of straight lines.

Take the underlying object to be the real number line:

We construct a sheaf by building not on individual points, but on intervals. You can break up the number line into intervals in infinitely many ways. One example is shown below.

Between each pair of matching parentheses is an interval that includes all the points between them, but not the endpoints. So the interval (0, 1) contains all the numbers that are bigger than zero and smaller than 1.

The sheaf contains all the intervals, not just any given one. Each interval can be assigned a set of “sections.” In this example, the sections are all possible straight lines that pass through an interval.

Take just a single interval, as shown below. Only three of the sections are shown, since it’s impossible to visualize all of them at once.

The sheaf comprises all sections on all possible intervals and unions of intervals.

This is a bewilderingly chaotic entity. It becomes mathematically intriguing because it conceals an underlying simplicity. In the figure above, the sections chosen for different intervals clash. Lines pass above and below each other, rather than coinciding.

Mathematicians are interested in understanding what happens when you choose one section from each interval and impose the requirement that different sections be compatible with each other, so that overlapping intervals agree. With that constraint, something remarkable happens.

If one interval is nested inside another, the lines have to match on the overlap.

From this local constraint, you get a global consequence. Instead of lots of little lines, you end up with the only possible choices that comply with the nesting rule: straight lines that continue over the whole number line.

These are called global sections. One thing that gives sheaves their power is that such global objects emerge from local constraints.

That’s a tour of the sheaf of straight lines, or linear functions, over the real line. It’s one of the simplest sheaves.

You can create lots of sheaves over the real line. This is analogous to planting different flowers in a garden on the same plot of land. There’s a sheaf consisting of functions whose graphs have no jumps, a sheaf of functions whose graphs have no sharp corners, and infinitely many others.

But that’s only the beginning. Instead of planting a different flower, you could tend to a different plot of land. Imagine building a sheaf on top of a circle, rather than a line. This creates a structure that looks like a cylinder with infinite height. The structure of the objects drawn on that cylinder depends on the particular construction for a specific sheaf.

Up to this point, all the sheaves we’ve considered can be thought of as families of functions. But sheaves can get (much) more complicated than that.

The cylinder in the figure above can be thought of as coming from an infinitely tall rectangle whose sides you’ve glued together. If, instead, you were to twist the ends of the rectangle before gluing them, as in the figure below, you’d create an infinitely wide Möbius strip (it’s not possible to draw this, so we’ll show a finite Möbius strip). On this Möbius strip, you can still draw curves reminiscent of graphs.

On any little local piece of the circle, this curve looks like the graph of a function. But on a global scale, it’s not a function. That’s because there is no way to define a consistent global coordinate system, because of the twist. (If you travel all the way around the strip, your notions of up and down end up flipping, making it impossible to do so.) Mathematicians call such objects “twisted functions.”

While each sheaf is a vast collection of objects, you can also consider the collection of all sheaves on a given mathematical object — the real line, a circle, or some other entity. This is like considering all the possible gardens you could plant on a given plot of land. This tells you something about what that land is like. Some plots are rainforests, others are deserts. Figuring out what sheaves are possible gives mathematicians a way to probe the structure of the underlying space, just as knowing which plants grow in a particular type of soil gives you information about that soil.

Starting with Grothendieck, mathematicians gradually realized that collections of sheaves have many commonalities with collections of functions, but at a higher level of complexity. You can add and multiply sheaves, and even do a version of calculus on them.

In prison, Leray had opened the door to a whole new mathematical world.