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Which of the results mentioned in the article are proven by applying some result of category theory to another field, and which are just stated using category t

Which of the results mentioned in the article are proven by applying some result of category theory to another field, and which are just stated using category theory (with the real mathematical insight lying elsewhere)?

whatshisface 4y ago
If you translate everything to being stated in terms of category theory, it clarifies how it all works and makes it easier for people to go from one area to another. It is a bit like what Grothendieck did with abstract algebra. Imagine if every sub-field of mathematics had their own word for group, and that's something like how category theory fans see the present world. There's no such thing as a theorem that can only be proven with category theory, because whatever specific thing you're applying category theory to already has all of the properties that would make the theorem true without category theory being involved - if it didn't, you wouldn't be able to treat it as if it was a category. Categories are not the kind of mathematical invention that, for example, ordinary differential equations or matrices are.
- cwzwarich 4y ago
  You could say the same thing for any abstract mathematical construct, e.g. metric spaces. There's no theorem (that's not itself about metric spaces) that can only be proven using metric spaces. It's always possible to make new abstractions (simply take the antecedent of any theorem), so whether you adopt a new abstraction is a matter of taste and fashion. In Grothendieck's early work on abelian categories, categories are themselves the objects of mathematical study, and nontrivial results about them are established.
- isitmadeofglass 4y ago
  > If you translate everything to being stated in terms of category theory, it … … makes it incomprehensible to experts in the field and laymen alike!
- hackandthink 4y ago
  I agree: "There's no such thing as a theorem that can only be proven with category theory". But sometimes it feels like Category Theory is rebuilding the world and there are genuine theorems. Epilogue: Theorems in Category Theory https://emilyriehl.github.io/files/context.pdf https://emilyriehl.github.io/files/context.pdf

whatshisface 4y ago

If you translate everything to being stated in terms of category theory, it clarifies how it all works and makes it easier for people to go from one area to another. It is a bit like what Grothendieck did with abstract algebra. Imagine if every sub-field of mathematics had their own word for group, and that's something like how category theory fans see the present world. There's no such thing as a theorem that can only be proven with category theory, because whatever specific thing you're applying category theory to already has all of the properties that would make the theorem true without category theory being involved - if it didn't, you wouldn't be able to treat it as if it was a category. Categories are not the kind of mathematical invention that, for example, ordinary differential equations or matrices are.

cwzwarich 4y ago
You could say the same thing for any abstract mathematical construct, e.g. metric spaces. There's no theorem (that's not itself about metric spaces) that can only be proven using metric spaces. It's always possible to make new abstractions (simply take the antecedent of any theorem), so whether you adopt a new abstraction is a matter of taste and fashion. In Grothendieck's early work on abelian categories, categories are themselves the objects of mathematical study, and nontrivial results about them are established.
isitmadeofglass 4y ago
> If you translate everything to being stated in terms of category theory, it … … makes it incomprehensible to experts in the field and laymen alike!
hackandthink 4y ago
I agree: "There's no such thing as a theorem that can only be proven with category theory". But sometimes it feels like Category Theory is rebuilding the world and there are genuine theorems. Epilogue: Theorems in Category Theory https://emilyriehl.github.io/files/context.pdf https://emilyriehl.github.io/files/context.pdf