My next series of posts will be directly about the Yoneda lemma, which basically tells us that everything you could want to know about an object is contained in the morphisms going into/out of the object. Moreover, we get this knowledge in a “natural” way that makes life really easy. It’s a pretty cool theorem.
In the end, we don’t really care about sets at all. They’re just bags with stuff in them. Who cares about bags? But we do care about functions—we want those to be rule-based. We need functions to go “from” somewhere and “to” somewhere. Let’s call those things sets. Then we need these “sets” to be rule-based.
I’m grateful for your comments. They’re very useful, and you raise good points. I’ve got most of a post already about how functions give meaning to the elements of sets. As for how functions is a verb, think of properties as existing in verbs. So to know something, you need to observe it in some way, which means it has to affect your sensory devices, such as your ears, eyes, thermometers, whatever. You know dogs, for example, by the way they bark, by the way they lick, they way they look, etc. So properties exist in the verbs. “Legs” are a noun, but all of your knowledge about them has to come from verbs. Does that make sense?
You raise a good point. Think of category theory as a language for expressing, in this case, the logic of sets and functions. You still need to know what that logic is. Then you can use category theory to work efficiently with that logic owing to its general-abstract nature.
I agree. I’m shifting gears to work on something basically aimed at the idea that the intelligent layperson can grasp Yoneda lemma and adjunction if it’s explained.
It’s older terminology. Everyone says image now.
I thought I had it right, and then mixed it up in my head myself.
That was my intention. Thanks for pointing it out. One of the mistakes of this series was the naive belief that simplicity comes from vagueness, when it actually comes from precision. Dumb of me.
Steam is run out of. This was poorly conceived to begin with, arrogant in its inherent design, and even I don’t have the patience for it anyway. I’ll do a series about adjunction directly and Yoneda as well.
Honestly my real justification would be “adjoint functors awesome, and you need categories to do adjoint functors, so use categories.” More broadly...as long as it’s free to create a category out of whatever you’re studying, there’s clearly no harm. The question is whether anything’s lost by treating the subject as a category, and while I fully expect that there are entire universes of mathematics and reality out there where categories are harmful, I don’t think we live in one like that. Categories may not capture everything you can think of, but they can capture so much that I’d be stunned if they didn’t yield amazing fruit eventually. I’d acknowledge that novel, groundbreaking theorems are still forthcoming.
Thank you for the positive feedback. (A very underrated thing in terms of encouraging free content production.) I can go back to each post and add a link to the next one. I am concerned that I may want to add, rearrange, or even delete individual posts at some point, but I suppose that’s no reason not to add in the links right now for convenience’s sake.
One of the reasons for my own interest in category theory is my interest in the question you raise. I’m hoping that we’ll explore the idea that universal properties offer an “objective” way of defining “subjective” categories.
Maybe a more direct answer is that in the very next post in the series, we’ll see that sets can be considered the objects of the category of sets and functions, and also the objects of the category of sets and binary relations. Functions are binary relations, so that’s not a perfect answer, but yes, you can think of an individual category as a context of sorts through which you view the objects, like how you can view a tomato as a fruit or vegetable depending on the context.
You have thought about the language analogy much harder than I did. I will think about how to avoid this issue better in the future, so thank you. In any case, don’t stress it too much—all that this post seeks to establish is that category theory is a mathematics of “stuff taking action on stuff”—moreover, it does so in a logical, intuitive way that you are already familiar with, even if you don’t know higher maths. Judging by Said’s comment, I also should have clarified that specific branches of mathematics fill in particular things for “stuff” and “taking action.” E.g., you get set theory when you fill in “sets” for stuff and “functions” for taking action.
Thank you very much for your reaction to this post. As it happens, I find myself in agreement with you. I leaned too hard in the direction of avoiding any discussion of mathematics. The next post is already written to clarify that sentences are all about nouns and verbs because we use sentences to model reality, and reality seems to consist of nouns and verbs. (Cats, drinking, milk, etc., are all part of reality. Even adjectives like “blue” are broken down by our physics into nouns and verbs.) We use various specific kinds of mathematics to model various specific parts of reality, and so various specific kinds of mathematics themselves boil down to nouns and verbs. So when you do a “mathematics of math” it ends up being a mathematics that is analogous to a mathematics of nouns and verbs, which get called objects and morphisms respectively. (We probably can’t carry this analogy forever—I don’t know that there’s a real-world language analogy to n-categories. But that won’t come up anyway.) I’ll very much look forward to your reaction to the next post, which motivates category theory as a general description of how you’d want to model pretty much anything in a universe of cause-and-effect, which correspondingly generalizes, almost as a byproduct, the mathematics any human is likely to invent.
There are many options for being clearer about objects and morphisms in this post, and I will consider them...I will also take pains to ensure it is not necessary to reconsider future posts for this particular mistake, thanks to you.
I’m motivated by the essays themselves. I believe this material should exist. It’s also good for my own understanding to write them.
When we get to adjunction, I’m hoping the utility of the series will start to become clear.
You’re playing chess. You’re white, so it’s your move first. It’s a big board, and there’s a lot of pieces, so you’re not quite sure what to do.
Luckily for you, there definitely exists a rule that tells you the best possible move to play for every given configuration of pieces—the rule that tells you the move that maximizes the probability of victory (or since draws exist and may be acceptable, the move that minimizes the probability of defeat. Or maximizes the points you gain, 1 for a win and 0.5 for a draw, over an infinite number of games against the opponent in question—whatever.).
But many of these configurations of pieces will have more than one possible move to play, so it’s not like this rule is just a given. You have to figure it out.
So what is a rule? It’s something that tells you what move to play in any and every given position. Two rules are equal when, for every possible position, they tell you to play the same move.
When two rules are equal, we just merge them into one rule—they’re literally the same, after all. So let’s consider the list of all unequal rules—rules that differ in at least one move recommendation for at least one position from every other rule.
How many of these rules are there? Mathematically speaking, the answer is “a super-huge amount that would literally cause your mind to explode if you tried to hold them all in your head.” After all, there is a universe-eating number of possible chess positions (remember, this is a rule that works for all possible chess positions, even ones that would never happen in a real game). And in every chess position, the number of possible ways that rules can be distinguished is equal to the number of possible moves in that position.
So imagine each rule as a black ball, each attached to this really big wall in this vast infinite array. Out of this huge infinity of black balls, one of these balls gets a little pink dot placed on its backside, so you can’t see that it’s there.
Now, out of this huge infinite array, you have to find the one ball with the little pink dot on it. That is the challenge of finding the rule that tells you the best chess move for each position.
Is ethics just as hard? No! Ethics is insanely harder. Because ethics tells you a most ethical move possible for each chess position, which includes illegal moves and therefore there are more black balls corresponding to just the subset of ethical choices for chess positions!
Now consider that also Go exists, and checkers, and, Fortnite, and situations that aren’t even games at all, like most of everything in the universe.
There’s a ball for each way a rule can be distinguished from all other rules over the full list of all possible situations, including but hardly limited to chess situations.
One of them has a little pink dot on its backside. Go find it.
That is the challenge of ethics.
Yes, people disagree about which ball has the little pink dot on it! Yes, you can search your heart and soul and still not know which ball has the little pink dot on it! That does not mean there is not a ball with a little pink dot on it!
The pink-dotted ball exists!
Alas, the Babyeaters were looking for the ball with the little red dot on it, and the Super Happy people looking for the ball with the little blue dot on it. Looking for different colored dots, or disagreeing about which ball has the pink/red/blue dot on it, is the stuff that wars and trade are made of.