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[1] https://arxiv.org/pdf/1612.09375
Category theory is also not that impressive unless you already understand some of the semantics it is trying to unify. In this regards, the book itself presents, for example, the initial property as trivial at first hand, unless you notice that it does not simply hold for arbitrary structures.
[1, 3, 2].sort((a, b) => { if (a > b) { return true
})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
And to tie it down to the mathematics: if a sorting algorithm asks for a full comparison between a and b, and your function returns only a bool, you are conflating the "no" (a is before b) with the "no" (a is the same as b). This fails to represent equality as a separate case, which is exactly the kind of imprecision the author should be trying to teach against.
Let's scroll up a little bit and read from the section you're finding fault with:
Rather than the usual "harrumph! This writer knows NOTHING of mathematics and has no business writing about it," maybe a simple counter-example would do, i.e. present an ordering "in which every object has its place depending on every other object" and "leaves no room for ambiguity in terms of which element comes before which" but also satisfies your requirement of allowing 'equal' ordering.Ah! You're talking about Racket or Scheme!
```
> (sort '(3 1 2) (lambda (a b) (< a b)))
'(1,2,3)
```
I suppose you ought to go and tell the r6rs standardisation team that a HN user vehemently disagrees with their api: https://www.r6rs.org/document/lib-html-5.96/r6rs-lib-Z-H-5.h...
To address your actual pedantry, clearly you have some implicit normative belief about how a book about category theory should be written. That's cool, but this book has clearly chosen another approach, and appears to be clear and well explained enough to give a light introduction to category theory.
Nobody seems to care or notice. I'm watching in disbelief how nobody is pointing out the article is full of inaccuracies. See my sibling thread for a (very) incomplete list, which should disqualified this as a serious reading: https://news.ycombinator.com/item?id=47814213
My conclusion cannot be other than this ought to be useless for the general practitioner, since even wrong mathematics is appreciated the same as correct mathematics.
https://math.stackexchange.com/questions/823289/abstract-non...
Sometimes the proof in category theory is trivial but we have no lower dimension or concrete intuition as to why that is true. This whole state of affairs is called abstract nonsense.
Writing a program and proving a theorem are the same act. (Curry–Howard–Lambek.) For well-behaved programs, every program is a proof of something and every proof is a program. The match is exact for simple typed languages and leaks a bit once you add general recursion (an infinite loop “proves” anything in Haskell), but the underlying identity is real. Lambek added the third leg: these are also morphisms in a category. [1]
Algebra and geometry are one thing wearing different costumes. (Stone duality and cousins.) A system of equations and the shape it cuts out aren’t related, they’re the same object seen from opposite sides. Grothendieck rebuilt algebraic geometry on this idea, with schemes (so you can do geometry on the integers themselves) and étale cohomology (topological invariants for shapes with no actual topology). His student Deligne used that machinery to settle the Weil conjectures in 1974. Wiles’s Fermat proof lives in the same world, though it leans on much more than the categorical foundations. [2]
[0] https://en.wikipedia.org/wiki/Yoneda_lemma
[1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
[2] https://en.wikipedia.org/wiki/Stone_duality
We should strive to name all things by their function not by their inventor or discoverer IMO. But people like their ribbons.
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.
I'm unclear what the last 10% of 'category theory' gives us.
In a preorder seen as a category, there is at most one arrow between any two objects. So every diagram commutes and uniqueness is basically free. Then products and coproducts stop looking like magic diagrams and become something very familiar: greatest lower bounds and least upper bounds.
Small nit: preorders are thin categories, but posets are the skeletal thin categories. In a preorder you can have distinct a and b with both a <= b and b <= a, which means they are isomorphic, not literally the same. Quotienting by that equivalence gives you the poset.
The software angle is the part I find most useful. This kind of bugs shows up when we force a total order onto something that is only partially ordered, or only preordered. Dependency graphs, versions, permissions, type hierarchies, CRDT states, rule specificity, build steps. A lot of these don’t really want a comparator and a sort. Sometimes they want a quotient, a topological sort, a join, or just the honest answer that two things are not comparable.
That feels like the practical lesson here: category theory is not always adding abstraction. Sometimes it is just a good way to stop pretending two different structures are the same thing.
A morphism from orange to yellow means "O <= Y". From this, antisymmetry (and the hidden assumption) implies that "Y not <= O".
Totality is just the other way around (all two distinct elements are comparable in one direction).
"All diagrams that look something different than the said chain diagram represent partial orders"
"The different linear orders that make up the partial order are called chains"
The Birkhoff theorem statement, which is materially wrong. A finite distributive lattice is not isomorphic to "the inclusion order of its join-irreducible elements".
The 'not accurate' diagram says that orange-less-than-yellow implies yellow-not-less-than-orange. Hard to find fault with.
> 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.
I like the article's "imprecise prose" better:
The prose "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" is inaccurate for describing antisymmetry. In the same short section, you first state the correct condition:
You have x ≤ y and y ≤ x only if x = y
from which it doesn't follow that "It also means that no ties are permitted". The "no ties" idea belongs to a stronger notion such as a strict total order, not to antisymmetry.