Category Theory Illustrated – Orders

If you want to learn category theory in a way that is more orthodox, a lot of people recommend Tom Leinster’s Basic Category Theory, which is free[1]. I’m going to be working through it soon, but the bit I’ve skimmed through looks really good if more “mathsy” than things like TFA. It also does a better job (imo) of justifying the existence of category theory as a field of study.

[1] https://arxiv.org/pdf/1612.09375

If someone does not want to check the mathematics line by line and prefers to give the article the benefit of the doubt, note that it also presents this JavaScript:

[1, 3, 2].sort((a, b) => { if (a > b) { return true

  } else {

    return false
  } 

This is not a valid comparator. It returns bools where the API expects a negative, zero or positive result, on my Chrome instance it returns `[1, 3, 2]`. That is roughly the level of correctness of the mathematics in the article as well, which I'm trying to present in sibling comment: https://news.ycombinator.com/item?id=47814213

I think it is pretty obvious that at the challenge with all abstract mathematics in general and the category theory in particular isnt the fact that people dont understand what a "linear order" is, but the fact it is so distant from daily routine that it seems completely pointless. It's like pouring water over pefectly smooth glass

There is a way to frame category theory such that it's all just arrows -- by associating the identity arrow (which all objects have by definition) with the object itself. In a sense, the object is syntactic sugar.

This resource is a really clear breakdown of order relations; visualizing the structure like this makes the abstract concepts much more digestible

binary relations defining order are more nuanced than they seem; a linear order isn't just about ranking, it's about the structure of the relationships themselves.

studying category theory for my master's in 2015 showed me how orders influence everything from data structures to algorithms. foundational stuff.

this reminds me of Haskell’s type classes; they elegantly define order concepts through their own set of rules, capturing relationships in a clean way.

I love how math is like a new language, in a new country, of culture you are not exactly familiar with.

This article is like living there for few months. You see things, some of them you recognize as something similar to what you have at home, then you learn how the locals look at them and call them. And suddenly you can understand what somebody means when they say:

"Each distributive lattice is isomorphic to an inclusion order of its join-irreducible elements."

Having a charitable local (or expat with years there under their belt) that helps you grasp it because they know where you came from, just like the person who wrote this article, is such a treasure.

The first 90% of this is standard set theory.

I'm unclear what the last 10% of 'category theory' gives us.

Unless there's some idiosyncratic meaning for the `=>`, the Antisymmetry one basically says `Orange -> Yellow => Yellow -/> Orange`. The diagram is not acurate. The prose is very imprecise. "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me." NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.