Category Theory Illustrated – Orders
abuseofnotation.github.io
Category Theory Illustrated – Orders
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Re: Category Theory Illustrated – Orders
#2There 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.
Re: Category Theory Illustrated – Orders
#3I 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
Re: Category Theory Illustrated – Orders
#4I 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?
Re: Category Theory Illustrated – Orders
#5[deleted]
Re: Category Theory Illustrated – Orders
#6I 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?
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?
Re: Category Theory Illustrated – Orders
#7Unless 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.
Re: Category Theory Illustrated – Orders
#8I 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?
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 different rings can have the same modules, for instance.) [0]
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...
Re: Category Theory Illustrated – Orders
#9Unless 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).
Re: Category Theory Illustrated – Orders
#10Earlier 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…
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