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Category Theory Illustrated – Orders

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Re: Category Theory Illustrated – Orders

#11

Post by gobdovan »

bubblyworld wrote: post #9
gobdovan wrote: post #7

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 no…

I don't think they are completely wrong - "=>" is just implication. A hidden assumption in their diagrams is that circles of different colours are assumed to be different elements. A morphism from orange to yellow means "O Totality is just the other way around (all two distinct elements are comparable in one direction).

If this is meant to be an explainer, that can't be simply implicit. The text actually seems full of imprecise claims, such as:

"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".

Re: Category Theory Illustrated – Orders

#13

Post by throw567643u8 »

raincole wrote: post #4
dgan wrote: post #3

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

Is there a "mind-blowing fact" about category theory? Like the first time I've heard that one can prove there is no analytical solution for a polynomial equation with a degree > 5 with group theory , it was mind-blowing. What's the counterpart of category theory?

Just Yoneda Lemma. In fact it feels like the theory just restates Yoneda Lemma over and over in different ways.

Re: Category Theory Illustrated – Orders

#15

Post by wallhz »

U4E4 wrote: post #8
raincole wrote: post #4

Earlier quoted context omitted.

Is there a "mind-blowing fact" about category theory? Like the first time I've heard that one can prove there is no analytical solution for a polynomial equation with a degree > 5 with group theory , it was mind-blowing. What's the counterpart of category theory?

A thing is its relationships. (Yoneda lemma.) Keep track of how an object connects to everything else, and you’ve recovered the object itself, up to isomorphism. It’s why mathematicians study things by probing them: a group by its actions, a space by the maps into it, a scheme in algebraic geometry defined as the rule for what maps into it look like. (You do need the full pattern of connections, not just a list — two…

[dead]

Re: Category Theory Illustrated – Orders

#16

Post by [deleted] »

raincole wrote: post #4
dgan wrote: post #3

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

Is there a "mind-blowing fact" about category theory? Like the first time I've heard that one can prove there is no analytical solution for a polynomial equation with a degree > 5 with group theory , it was mind-blowing. What's the counterpart of category theory?

[deleted]

Re: Category Theory Illustrated – Orders

#17

Post by gobdovan »

dgan wrote: post #3

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

You're more right than you'd think. The whole point of mathematics is precise thinking, yet the article is very inaccurate.

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.

Re: Category Theory Illustrated – Orders

#18

Post by brador »

U4E4 wrote: post #8
raincole wrote: post #4

Earlier quoted context omitted.

Is there a "mind-blowing fact" about category theory? Like the first time I've heard that one can prove there is no analytical solution for a polynomial equation with a degree > 5 with group theory , it was mind-blowing. What's the counterpart of category theory?

A thing is its relationships. (Yoneda lemma.) Keep track of how an object connects to everything else, and you’ve recovered the object itself, up to isomorphism. It’s why mathematicians study things by probing them: a group by its actions, a space by the maps into it, a scheme in algebraic geometry defined as the rule for what maps into it look like. (You do need the full pattern of connections, not just a list — two…

We should call it “relationship lemma”. That way its function is contained within its name. And would not require the definition step every time.

We should strive to name all things by their function not by their inventor or discoverer IMO. But people like their ribbons.

Re: Category Theory Illustrated – Orders

#19

Post by auggierose »

tux3 wrote: post #6
raincole wrote: post #4

Earlier quoted context omitted.

Is there a "mind-blowing fact" about category theory? Like the first time I've heard that one can prove there is no analytical solution for a polynomial equation with a degree > 5 with group theory , it was mind-blowing. What's the counterpart of category theory?

Sure, category theory can't prove the unsolvability of the quintic. But did you know that a monad is really just a monoid object in the monoidal category of endofunctors on the category of types of your favorite language?

Phil?

Re: Category Theory Illustrated – Orders

#20

Post by mrkeen »

gobdovan wrote: post #7

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 no…

It really isn't a long enough section to get lost in.

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:

  You have x ≤ y and y ≤ x only if x = y
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