website/content/posts/fp.md

title	date	modified	draft	summary
Dysfunctional Programming	2025-11-24	2025-11-24	true	Group Theory Adjacent Nonsense for people who slept through high school (me)

Group Theory Adjacent Nonsense for people who slept through high school (me)

A Cringe set is Just a Co-Based set

- Unknown

So anyway, what the FUCK is a Monad

• A monoid in the category of endofunctors
• a burrito
• a co-comonad

Do any of these mean anything? No, so lets start at the beginning

Numbers

• In the beginning, there was the empty set
• and then Zermelo and Fraenkel said "Let there be a power set of the empty set"
• ...which is a line of exploration that will get us nowhere productive for the scope of this work, and is probably something I'll talk about in the future.

Magmas

• The easiest Algebraic Structure to explain is The Magma
• Definition: A magma is a set M with an operation \cdot that sends a,b\in M \to a\cdot b \in M
• the tl;dr is, you've got a set of stuff, and you can shove any two of those together to make a third thing, as long as it's still in the set, we call this property closure
  • you're probably used to \cdot meaning multiplication, but that's a fake word that doesn't mean anything
  • Aside: '(\cdot)' is hard to read on its own, hereafter it will be referred to as multiplication or as bottom surgery 'the operation')
• The main example of this is the Free Magma, which is basically just a binary tree
  • In this case, 'multiplying' two binary trees just means concatenating them
  • Why is this called the Free Magma? I honestly have no clue, Free means something here but I haven't figured out what yet. This will not be the last time it comes up.

Monoids

• The next step we'll on the Algebraic Structure Skill Tree is the Semigroup, but that's less important to focus on here so I'll fold it into the Monoid Section
  • When you add Associativity to a Magma, you get a Semigroup
    • Associativity just means that a\cdot(b\cdot c) = (a\cdot b) \cdot c
    • A more concrete example of this is natural number addition, 2+(3+4) = (2+3)+4
      • The proof of this is trivial and left as an exercise to the reader
  • When you take a Semigroup and add an Identity to it, you get a Monoid
    • An identity is just something that you can combine with something and get that same something back out
    • The easiest way to think of this is that we need something that acts like '1' when you 'multiply' by it
      • If we go back to the example of adding numbers together, '0' acts as the identity since 0+x = x
        • Yes, this means that the '1' element of (\mathbb{N},+) is '0'
        • No, I'm not sorry for coming up with that absolutely dumb ass example
• The Free Monoid is strings with concatenation as the operation
  • Told you 'Free' would come up again
  • the Identity element here is '', or the empty string
  • This is closed since adding two strings together always results in a string
  • This is also associative because Trust Me Bro
• Another example of a Monoid is RGBA colors
  • The identity here is rgba(0,0,0,0) since adding clear to any color doesn't change it
  • Colors are closed under saturating addition since you can add two of them and get a color back*
    • *Actually I'm sure there's some sort of floating-point nonsense that makes this not true but IEEE 754 can go suck a log
    • actually you know what, lets pretend that #RRGGBBAA is a real notation that real people use, since it'll let us replace define multiplication as 8-bit saturating addition which is a much more well defined thing to do to colors
  • RGBA Color addition is associative because I said so
  • Absolutely unrelated, but I'm pretty sure OKLab isn't closed under the set of displayable colors, which feels cursed but is probably reasonable tbh

Categories

• The Magma -> Monoid skill tree (and the several parts of it that I left out), are a scuffed refresher on Group Theory
• Up next is Category Theory, which they did not cover in high school; As such I will be spewing a significant amount of hot air out of my ass.

Functors

A category is a collection of "objects" that are linked by "arrows"

- Wikipedia - Category

• ...FUCK, I was going to say that, unfortunately Wikipedia's definition is pretty spot on here.
• Okay but seriously

In mathematics, abstract nonsense is a nonderogatory term used mainly to describe category theory and homological algebra

- Paraphrased from Wikipedia - Abstract nonsense

• Okay forget finding a way to explain it to an audience of (uhhh whatever my audience is, I kinda try to write these at a level high school me could follow along with and not get bored, but I don't actually know what kind of masochist is reading this), I need to figure out how to explain this to me first
• I'm back from Wikipedia, and I've returned neither smarter nor wiser, as such we proceed in our endeavors with the rallying cry: "Fuck it, we ball"

Endofunctors

Monads

• Examples include: Future<T>, List<T>, Option<T>, Result<T,E> also logging
  • TODO: find someone who can explain how/why logging and file i/o are monadic

---

Questions Mira hasn't answered yet and should really get around to before publishing this

• Does free just mean the operation is concatenation?
  • https://en.wikipedia.org/wiki/Free_object
  • it's supposed to be the 'most generic'
• How does using monads deal with side effects, eg Logging, File IO
  • Do they encapsulate state somehow? how does that work exactly?
• what exactly does Co- entail