Finally, a very concrete explanation of sheaves.

Recently, I’ve been fascinated by ~~sheaves~~, the most exciting and useful of tools to study algebraic geometry and plenty of other areas of math. Unfortunately, I’ve had immense difficulty finding a very introductory example of what a sheaf is and just some basic examples so I can wrap my head around the idea. In any area of math, I always think it’s good to have a very concrete example to start with, and from there you can dive into more abstractions while referring back to an easily-graspable example to build your intuition. And once you’ve built up an understanding of how these concepts are all tied together you can let go of this very basic notion and see these ideas for what they really are. Kinda like how jellyfish eggs settle to the ocean floor and grow into a plant-like thing, which then makes a bud and the bud grows and eventually pops off and is an adult jellyfish leaving it’s nursery-plant-home behind as it goes on to explore the ocean. Here is my attempt to give you the most basic, concrete, run-of-the-mill, vanilla, example of a sheaf.

I’m going to abuse a lot of notation, terminology, and conventions. This isn’t a chapter from a textbook, this is just to get our heads wrapped around the idea of what a sheaf is.

In general, a sheaf is an object that tells you information about a topological space by mapping subsets of the space to their own algebraic structures. Let’s start with a topological space. It could be anything, really. The real numbers \mathbb{R} are a great place to start. A set becomes a topological space when you devise a way to call some subsets “open” and that collection of open sets follow certain rules. Let X be a set and let \tau be a collection of subsets of X. Here are the rules:

  1. X,\emptyset \in \tau. Meaning that the entire set and the empty set are both labeled as open sets.
  2. Infinite unions of open sets are open. So we can take as many open sets as we want and take their union and the result will also be an open set.
  3. Finite intersections of open sets are open. So for any finite collection of open sets, their intersection is also open. Their intersection can even be empty and this still holds because the empty set is also an open set.

If you take any set and can think of a collection of subsets that follow these rules then your set becomes a topological space! Hooray!

So let’s look back at \mathbb{R}. The open sets are exactly the open intervals (this is where the generalized name for “open sets” came from). So (-3,1) is open, and so is (5,1201893483526). Even (-\infty, 12) and (100, \infty) are open. And any union of open sets are open and all the finite intersections too. It is also helpful to notice that given one set, say \mathbb{R}, you can find more than one way label sets as open. For instance, if our collection of open sets of \mathbb{R} is just \{\mathbb{R},\emptyset\} then this still satisfies the rules mentioned above, it’s just not very interesting. This is a very important thing to make note of, the fact that we can create different topologies on the same space.

Ok, we have topological spaces, now let’s introduce some algebraic objects. Then, a sheaf will be a way to go between them. very cool.

A ring is a set with two operations. We can think about the integers \mathbb{Z} as a ring. The two operations are addition (+) and multiplication (*). Rings also come with some rules that distinguish them from just sets. Let R be our ring. Here are the rules:

  1. for any two elements a,b \in R their sum and product are also in R. So for our example, the sum of two integers is a integer and the product of two integers is an integer.
  2. There is some element e \in R called the “additive identity” which does not change any elements additively. Which in our case is 0, because any number plus 0 is just itself.
  3. Each element has an “additive inverse” that when added together gives the additive identity. Which in our case is just the negative. So for the number -4, the additive inverse is -(-4) = 4 and we have -4 + 4 = 0.
  4. The distributive law holds. So for a,b,c \in R then a*(b+c) = (a*b)+(a*c). Which is nice.

Basically, the integers are a great example of a ring. So any property of the integers you can think of is probably true of all rings or at least most of them. But let’s use this idea of a ring to introduce another ring that will be helpful later.

The ring of continuous functions on \mathbb{R}

If you remember from grade school, continuous functions are the ones you can draw without lifting up your pencil. So the function f(x) = sin(x) is continuous, but the function f(x) = \frac{1}{x} is not because there is that jump as x=0.

We have a set, the continuous functions, and all we need are two operations that satisfy those rules mentioned above and our meager set will become a very pretty ring. So let “addition” be addition pointwise and “multiplication” be multiplication pointwise. Let’s get two continuous functions and illustrate this. Take f(x) = sin(x) and g(x) = x.

f(x) = sin(x) showing in red, and g(x) = x shown in blue. We can see that when we add them pointwise we get a continuous function (purple) and when we multiply them pointwise we also get a continuous function (black).

And the other rules apply as well. You can check if you don’t believe me. And boom! we have a ring. Now for the fun part.

~ s h e a v e s ~

Let \mathbb{R} be our topological space. Our sheaf \mathcal{F} will associate each open set of \mathbb{R} to its own ring. It’s actually pretty easy. For an open set U of the real numbers, let \mathcal{F}(U) be the ring of continuous functions on U. Below are a few elements of the ring \mathcal{F}(U) when U = (2,4).

But! like all things in math, there are a few rules that \mathcal{F} must follow in order to earn the prestigious title of sheaf. Here they are:

  1. If V \subset U are open sets of \mathbb{R} then there is a map from \mathcal{F}(V) to \mathcal{F}(U) by restricting each continuous function on U to V. Illustrated below. These next two are a bit tricky so bear with me.
  2. If f,g \in \mathcal{F}(U) and for all open subsets V \subset U, if f(x) = g(x) for all x \in V then f(x) = g(x) for all x \in U.
  3. This one I’m just going to do in English: If there is a set of functions in \mathcal{F}(U) that agree on all overlaps of the subsets of U, meaning that two functions are equal on all points of the intersection of two subsets of U, then there is one unique function in \mathcal{F}(U) that is equal to all of the overlapped functions glued together. I’ll explain with pictures below.
Rule 1: The subset (2.2,3.3) is a subset of (2,4). And so any continuous function on (2,4) can be restricted to the subset (2.2,3.3) and remain a continuous function on (2.2,3.3). It seems kinda obvious but that’s just because we are very familiar with this functions on the real numbers.

The last two are a lot to unpack, so I tried my best to draw out a guide as to what they mean.

Rule 2

Rule 2 states that if two functions are equal at all points on all open subsets of the larger subset in question, then they must be same function over the set. So here, f and g take the same values at all points in our set U so f=g in \mathcal{F}(U). Note that even if they are different outside of U, they are still considered to be the same element in \mathcal{F}(U).

Rule 3

This one is trickier. If there are a bunch of function in \mathcal{F}(U), I’ve labeled them 1 through 7 but have only colored them on certain open subsets of U, and every pair of these functions take the same values on intersections of these open sets, see how the red and blue functions overlap and so do the blue and green functions etc., then there is one function, the dotted black line, in \mathcal{F}(U) that is equal to all the other ones glued together. Oof that was a lot.

Hooray! Now our lovely \mathcal{F} has earned its name and is included in the world of sheaves.

Let’s wrap things up.

Hopefully, this gave you a brief, and very concrete introduction to sheaves. Now maybe when/if you go off to read up more on sheaves and they start throwing weird words and conditions and functors (ugh) at you, you can at least have a model of what a sheaf kind of looks like and from this example you can extend your understanding of sheaves in general.

I plan on doing more of these intros to mathematical and scientific ideas. My goal is to simplify the topics enough so that after you’ve read the post you have enough of a grasp on the topic that investigating it further isn’t so scary.

Hooray!

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