Products in the Category of Finite Sets: Their Non-Uniqueness

Consider the sets:

A = {a, b, c, d}
B = {b}.

Most people will tell you this is their product, when equipped with the "obvious" projections:

{⟨a,b⟩, ⟨b,b⟩, ⟨c,b⟩, ⟨d,b⟩}.

And so indeed it is. But it is a product, not the product. So let's see a different one for a change, one the books never show you:

P = {coffee cup, old penny, rubber dog, sock}.

Like all products, of course, it comes equipped with projection morphisms:

π1:P→A = {coffee cup→d, old penny→b, rubber dog→c, sock→a}
π2:P→B = {coffee cup→b, old penny→b, rubber dog→b, sock→b}.

Here is the isomorphism from it to the "usual" product:

{coffee cup→⟨d,b⟩, old penny→⟨b,b⟩, rubber dog→⟨c,b⟩, sock→⟨a,b⟩}.

Here is a diagram of this:

[diagram in VRML]

Now, I want to introduce another way to visualise this. In part, I hope that this will, as psychologists say, help you "chunk" it.

An important point about the categorical view of the world is that what is important is not the objects. It's their relation to other objects.

Here is a quote that puts forward this view, from philosopher David Corfield's book Towards a Philosophy of Real Mathematics, quoted in Quantum Quandaries: A Category-Theoretic Perspective by mathematical physicist John Baez:

Category theory allows you to work on structures without the need first to pulverise them into set theoretic dust. To give an example from the field of architecture, when studying Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don't do is begin by imagining it reduced to a pile of mineral fragments.

To visualise this, imagine that you are alone in an almost featureless, but brightly lit, void. You are standing on an inflated rubbery disc, about the size of a merry-go-round floor. Hanging in the void, slightly below your level and each about 100 yards away, are two other discs the same size as yours. On each is a flagpole with a flag waving in the breeze: one flag has a big yellow letter A; the other, a big yellow letter B.

From the edge of your disc run two U-shaped tubes or canals, like shiny black drainpipes cut in half. Both tubes have arrow heads painted on them, like a sequence of sergeant's chevrons, all pointing down the tube away from you. One tube runs down to disc A, the other to disc B. And if you had a rubber ball, you could bend down and roll it into the end of a tube, then watch it roll right down to disc A or B.

You don't have a rubber ball, but you do have a pile of other things at your feet. These are {coffee cup, old penny, rubber dog, sock}. In other words, the set P.

As you're looking at these, you stub your little toe on the coffee cup. It hurts, and in disgust, you grab the coffee cup and hurl it down the first tube, to disc A. Now, this is a magic tube!. As you roll objects down it, they transform. In this case, the coffee cup magically transforms into the corresponding element of our set A. Which is d.

Similarly, if you roll the coffee cup down the second tube to disc B, it magically transforms into the corresponding element of our set B. Which is b.

The point is that the essence of the product P is entirely carried by its relation to the other objects. And that relation is what you see as you stand on P (or any other object) and stare around you, looking at the tubes or arrows running into your disc and out of it; into other discs and out of them. Arrows are all around you, above and below you, on either side of you, in front of and behind you, dominating your view like Saturn's rings, or the arches of Larry Niven's Ringworld, or those ubiquitous overhead walkways in 1950s paintings of the future.

So imagine standing now on another bouncy rubber disc, the original product I showed you. It also has two arrows running out of it to discs A and B. And at your feet is another pile of things, the elements of its set {⟨a,b⟩, ⟨b,b⟩, ⟨c,b⟩, ⟨d,b⟩}.

When you were on P, you rolled the coffee cup to A, and it changed into d. Here, the corresponding thing you roll to A is ⟨d,b⟩, and it changes into d. And similarly for the arrows that lead to B. The view from both products, the one you were on, and the one you are now on, when you take into account what the arrows are doing, is essentially the same. And so the two products are, in essence, the same.

But what's happening now? You're growing; or perhaps the discs and tubes are shrinking! The discs get smaller and smaller until they're no larger than 10p pieces. The tubes dwindle to flexible cables and turn transparent, and the whole thing floats in front of you, an intricate and elegant little mechanism about the size of a wall clock.

Only, a wall clock is almost all casing. What you see before you is almost all cables: almost all arrows. They make a pattern that trembles on the edge of intelligibility; for in this pattern of relationships resides the essence of the entire system.

You grab the original product in your left hand, and the product P in your right. There are two cables leading from each. You notice tiny bulges working their way down each cable, like gobbets of food working their way down soneone's gullet. And you see a tiny man standing on one product disc, and a tiny woman on the other, picking up elements of the appropriate set and ramming them down each cable. You bring the original product close up to your left eye, and the product P ' to your right. The man and the woman are pushing corresponding elements down the cables, and these elements are magically mutating into what they have to be for the target disc A or B. Man and woman are working in perfect sync; the man's cables and the woman's cables are performing transformations that are different, yet in some sense the same.'

So once again, if you judge by the arrows, the two products are, in essence, the same.

Note:
The metaphor of tubes and transformations, I took from Toby Bartels's Quantum Gravity Seminar.