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Many results in algebraic geometry about rational points on curves are impossible to prove without category theory. Here's a pop-sci kind of take on it from qua

Many results in algebraic geometry about rational points on curves are impossible to prove without category theory. Here's a pop-sci kind of take on it from quanta that I like. The "bridge" between areas of mathematics they talk about are functors between categories. Also many proofs of the theorems you encounter in an undergrad level abstract algebra course are much easier using categories. Also, just from the tautological stand-point, if you can prove something about multiple types of mathematical objects at once using category theory, then you don't need to prove it for each of those objects individually. https://www.quantamagazine.org/amazing-math-bridge-extended-beyond-fermats-last-theorem-20200406/ https://www.quantamagazine.org/amazing-math-bridge-extended-...

spekcular 6y ago
None of those results would be impossible to prove without category theory, and all are easily translatable to non-category theoretic language. The use of categories doesn't provide any substantive input, it's just a useful shorthand. (I might have to make an allowance for the use of derived categories here, but I think I could still pull it off by being sufficiently concrete.) Further, none of the proofs from an undergrad abstract algebra are going to be "easier" with categories. (Show me one?) At the end of the day, you have to prove the same propositions. Categories just give a different and sometimes more convenient language.
- WoahNoun 6y ago
  I'd love to see a proof of Fermat's or Mazur/Tate without using any category theory. I suppose you could define lifts between categories by being explicit about the morphisms and functors but that's just category theory without the name. As for the second part, it's not that one single proof or another is easier with category theory. But the entire presentation is very elegant. Thinking of Aluffi Chapter 0 where he uses universal property extensively to simplify a lot of proofs. My personal favorite presentation of Algebra, but everyone has an opinion here.
  - spekcular 6y ago
    Well yeah, that's my point re: the number theory. It's a linguistic gloss that doesn't do much substantive work. Hatcher's book on Algebraic Topology, to give a random example, is able to cover lifts perfectly well without any category theoretic terminology. I'm all for good notation but the hype on certain HN stories seems really out of control. Re: Aluffi, it's a fine book, but I'm not sure I'd teach it that way, because I'm not sure what introducing universal properties and so on really buys you at that level. I guess it make sense if you commit to getting to the end with the abstract perspective on homological algebra. But for a grad student just looking to pass his or her quals? The discussion in Hatcher is concrete, motivated, and involves zero category talk.
    - WoahNoun 6y ago
      I love Hatcher's book so I can't disagree with you. (Although he does have a section on category theory in chapter 2, but never revisits it again). But doing anything with sheaves and schemes without categories would feel a lot like doing linear algebra without matrix notation and just using lots of double sums. I also have a copy of JP May's a Concise course in Algebraic Topology next to me and I also love the presentation there. (Although I would absolutely never want to use that book as a first course haha. Hatcher is a classic for a reason.) Being slightly facetious here as obviously Griffiths and Harris uses sheaves extensively and no category theory.
      - spekcular 6y ago
        Yeah, I totally agree. It's just I see a lot of category hype on here and I want to yell that it's mostly just linguistic! It's not some deep or scary thing. The analogy to matrix notation is really good.
        WoahNoun 6y ago
        Oh yea, the computer science side of things seems to rename anything related to abstract algebra as "category theory."

spekcular 6y ago

None of those results would be impossible to prove without category theory, and all are easily translatable to non-category theoretic language. The use of categories doesn't provide any substantive input, it's just a useful shorthand. (I might have to make an allowance for the use of derived categories here, but I think I could still pull it off by being sufficiently concrete.) Further, none of the proofs from an undergrad abstract algebra are going to be "easier" with categories. (Show me one?) At the end of the day, you have to prove the same propositions. Categories just give a different and sometimes more convenient language.

WoahNoun 6y ago
I'd love to see a proof of Fermat's or Mazur/Tate without using any category theory. I suppose you could define lifts between categories by being explicit about the morphisms and functors but that's just category theory without the name. As for the second part, it's not that one single proof or another is easier with category theory. But the entire presentation is very elegant. Thinking of Aluffi Chapter 0 where he uses universal property extensively to simplify a lot of proofs. My personal favorite presentation of Algebra, but everyone has an opinion here.
- spekcular 6y ago
  Well yeah, that's my point re: the number theory. It's a linguistic gloss that doesn't do much substantive work. Hatcher's book on Algebraic Topology, to give a random example, is able to cover lifts perfectly well without any category theoretic terminology. I'm all for good notation but the hype on certain HN stories seems really out of control. Re: Aluffi, it's a fine book, but I'm not sure I'd teach it that way, because I'm not sure what introducing universal properties and so on really buys you at that level. I guess it make sense if you commit to getting to the end with the abstract perspective on homological algebra. But for a grad student just looking to pass his or her quals? The discussion in Hatcher is concrete, motivated, and involves zero category talk.
  - WoahNoun 6y ago
    I love Hatcher's book so I can't disagree with you. (Although he does have a section on category theory in chapter 2, but never revisits it again). But doing anything with sheaves and schemes without categories would feel a lot like doing linear algebra without matrix notation and just using lots of double sums. I also have a copy of JP May's a Concise course in Algebraic Topology next to me and I also love the presentation there. (Although I would absolutely never want to use that book as a first course haha. Hatcher is a classic for a reason.) Being slightly facetious here as obviously Griffiths and Harris uses sheaves extensively and no category theory.
    - spekcular 6y ago
      Yeah, I totally agree. It's just I see a lot of category hype on here and I want to yell that it's mostly just linguistic! It's not some deep or scary thing. The analogy to matrix notation is really good.
      - WoahNoun 6y ago
        Oh yea, the computer science side of things seems to rename anything related to abstract algebra as "category theory."