Here’s a category-theoretic perspective. (Check out the rest of the lectures and the associated free textbook.)
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.