From Nothing to Something: Discrete Categories

Reading “Conceptual Mathematics” by F. William Lawvere & Stephen H. Schanuel.

Relax and let your mind go blank…

Congratulations: you have just entered your first category.

‘Empty Mind’ is the primary goal of most meditation programs.

Buddhists call empty mind meditation Vipassana, or Anapanasati. They devote themselves to years, sometimes an entire lifetime, of dedicated practice to achieve the state where thoughts disappear completely or no longer arise.

If every category is a “mathematical universe”, then perhaps “the empty category” is the universe which Buddhists strive to enter? This category contains no objects, no arrows, nothing. It is a perfect void, hence its name: 0.

Once you are able to hold the state of no thought, it becomes possible to focus your mind indefinitely on one point.

Now imagine “zooming out” from the empty category, like zooming out in Google Earth, until you’ve left that category and moved out into empty space. Keep flying further and further away from it until, when you look back, it’s so tiny that it’s just an infinitesimally small point, like a distant star. You have now entered a new category: 1. This category contains a single object, namely the empty category which you just left. This object, like all objects in categories (and unlike stars), is featureless: it has no color, no size, no shape, no mass, no properties of any kind. It is a purely abstract thing—a point in the (conceptual) void.

The acts of the mind, wherein it exerts its power over simple ideas, are chiefly these three: 1. Combining several simple ideas into one compound one, and thus all complex ideas are made. 2. The second is bringing two ideas, whether simple or complex, together, and setting them by one another so as to take a view of them at once, without uniting them into one, by which it gets all its ideas of relations. 3. The third is separating them from all other ideas that accompany them in their real existence: this is called abstraction, and thus all its general ideas are made.

An Essay Concerning Human Understanding
John Locke

Now imagine leaving 1 and entering a new category that contains exactly two objects in it. We might choose to label them A and B, or we might just draw them as two unlabeled dots:

A category with two unrelated objects in it.

You might guess that this category is called 2, but you would be wrong. (That name is reserved for another, more interesting, theoretical model of “two-ness”.) What are these two objects? Perhaps they are the two categories 0 and 1 that we began with… or perhaps they are a banana and the Eiffel Tower? They can be anything your heart desires. All we know for sure is that there are two of them, and that they have no relationship to one-another.

You become able to enter higher meditations easily and remain in them.

Continuing in this way, we see that we can easily construct categories with any number of objects in them. They are not very interesting categories, because the objects have no relationship with one another. They are like a bunch of people who are each stuck in individual quarantine (obligatory COVID-19 reference: check), not interacting with each other in any way. In fact, if you drew a picture of one of these categories, e.g., one with three objects in it, it would look rather like the internal diagrams at the bottom of page 13 of Lawvere and Schanuel:

“But,” I hear you object, “these are pictures of sets. I thought we were imagining categories?” Yes. Yes we were.

A category is discrete when every arrow is an identity. Every set X is the set of objects of a discrete category (just add one identity arrow x→x for each xX), and every discrete category is so determined by its set of objects. Thus, discrete categories are sets.

Categories for the Working Mathematician, 2nd Edition
Saunders Mac Lane
Springer, 1979

So, if we don’t bother drawing the identity arrows (and, in practice, identity arrows are customarily omitted), these drawings of the set {John, Mary, Sam} can also be interpreted as representing discrete categories. As we learn at the end of Article I, every object X in a category has an identity arrow that goes from X to itself: XX. So the only category that has no arrows at all is the category that has no objects at all, namely the empty category, 0. But though all discrete categories (except 0) have arrows, they are all identity arrows, and identity arrows are the most boring kind of arrows, because they represent the most boring kind of relationship. An identity arrow XX pretty much just says X = X, which philosophically means something like, “any thing is itself”. This truth is so obvious that you might think it not worth mentioning, but identity arrows will have an important role to play later when we start exploring more interesting categories with non-identity arrows in them.

These three charmingly simple diagrams of the set {John, Mary, Sam} have another “blink-and-you’ll-miss-it” feature: they progress from a concrete set of three people, to three labeled dots, to three unlabeled dots. That may not seem very Earth-shattering, but this is a typical example of how this book sneakily gets you to ponder sophisticated ideas without having to use a lot of distracting vocabulary. In a more typical book, you’d probably get a long-winded explanation about “concrete” vs “abstract” notions; for example, “three eggs” and “three megabytes” are concrete numbers, while “three” and “3” are abstract numbers, the key difference being that an abstract number has no intended interpretation—i.e. the abstract number 3 represent “three things” such that we don’t really know anything about the things except that there are exactly three of them. It’s a concept so familiar we don’t even think about it anymore, but just as there are concrete numbers and abstract numbers, there are also concrete sets and abstract sets.

So instead of a dry, hard-to-follow explanation of what it means to be a concrete set vs an abstract set, we get a little comic strip that neatly illustrates it. In fact, the phrase “abstract set” is not in the Index, and does not even appear in the book until Session 4 (page 62), where it is quietly snuck-in there so subtly, you probably wouldn’t notice it on a first reading. It’s interesting to compare this book with a similar one that Lawvere co-authored, Sets for Mathematics, which dispenses with the cutesy examples and starts off straight-away talking about what it means to be an abstract set:

An abstract set is supposed to have elements, each of which has no structure, and is itself supposed to have no internal structure, except that the elements can be distinguished as equal or unequal, and to have no external structure except for the number of elements.

Sets for Mathematics
F. William Lawvere and Robert Rosebrugh
Cambridge University Press, 2010

This description (which is on the very first page of the book!) really threw me at first, because I wondered, “if the elements of an abstract set have no structure, how can you possibly distinguish them??” I puzzled over this for quite a while (days, in fact), before I finally realized that what is meant is that the elements have no mathematically-relevant structure. Obviously, if you want to form a concrete set of things, then those things must have some properties that allow us to distinguish them, otherwise we couldn’t tell them apart! The point here is that, when we abstract them further into “an abstract set of n things” (where n is a natural number like 0, 1, 2, 3, …), we forget any properties that those things originally had. The original things still had to satisfy Cantor’s criteria of being “definite” and “distinct”:

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

Beiträge zur Begründung der transfiniten Mengenlehre
Georg Cantor

So once we’ve abstracted them into an abstract set, they essentially become just “n things”. As Lawvere and Schanuel finally admit in Session 6 (p. 81), “The abstract sets we are talking about are only little more than numbers, but this little difference is enough to allow them to carry rich structures that numbers cannot carry.”

This has three interesting implications. One is that any two abstract sets with the same number of elements are “the same“, in a sense which we will be able to make precise once we get to Article II. Another is that “the category of sets and maps” is a (much less scary) name for “the category of discrete categories and functors”, because a discrete category is essentially just a set, and a functor (the thing that lets us pass from one category to another) from one discrete category to another is (in this case) just a mapping of its objects/elements. The third and final implication is that, as Lawvere and Rosebrugh point out in Sets for Mathematics, sets are the “case zero” of structure, because abstract sets (aka discrete categories) have no structure at all—they are just “bags of dots”. In Article III we will start looking at some simple, but powerful, notions of “structure” (ways that elements can stick together, relate to one another, or move from one state to another), and see how category theory deals with them.

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