6 ms·

> ultimate abstraction (category theory), and it cannot go more abstract than that. I’m curious. What makes you think that?

leshow 9y ago
Not the other poster, but CT is the language of composition and abstraction, it's the high level language of math. I think people are just butt-hurt that he used the word 'ultimate'.
- dsacco 9y ago
  Category theory is not “the high level language of math”. You cannot universally apply category theory to all other fields of mathematics.
  - leshow 9y ago
    What fields of math do you think it doesn't apply to?
    - dsacco 9y ago
      Analysis.
      - taktoa 9y ago
        Not true: http://www.math.cornell.edu/~oconnor/sia.pdf http://www.math.cornell.edu/~oconnor/sia.pdf https://ncatlab.org/nlab/show/synthetic+differential+geometry https://ncatlab.org/nlab/show/synthetic+differential+geometr...
        dsacco 9y ago
        You cannot build all of analysis from category theory. Categories, morphisms and functors are not nuanced enough to encapsulate everything in analysis. The question was not, “Can you restate some results in category theory”, the question was “what cannot category theory build?” What you’re linking to is part of a relatively small corpus outside of e.g. algebraic geometry and topology that has successfully been used for nontrivial results. And as another commenter said, while it can be used for e.g. computation, it is extremely poor for complexity theory. There is a reason why basically any analyst (as opposed to algebraic geometer) will use set theory instead of categories for their work.
proc0 9y ago
Because you cannot discard more detail than what CT has chosen to discard. You start with objects and arrows, and how the arrows compose. Every object has an identity which is another arrow, so in some sense you have just a composition of arrows, and that's it. CT starts really simple, and I don't think there's anything left to hide or abstract.
- Koshkin 9y ago
  You may be wrong here, there have been some discussions - see for example https://www.quora.com/Whats-more-general-than-category-theory https://www.quora.com/Whats-more-general-than-category-theor...
- mbid 9y ago
  Every category is an infinity category, but the converse is not true. Thus, infinity categories are more general than categories. But you don't have to go that far: Just remove e.g. identity morphisms from categories, and you get something that is more general than categories. It is of course another matter completely whether these structures are a useful abstraction for what you want to accomplish. I've never come across something reasonable that has arrows and associative composition, but no identities. As for infinity categories, I remain very skeptical that they abstract any problems programmers not concerned with topology (i.e. > 99%) have. So yeah, categories are a reasonable baseline from where you can build. But you can't show anything relevant and non-trivial that holds in any category. You'll need more structure, hence less generality, for that.

leshow 9y ago

Not the other poster, but CT is the language of composition and abstraction, it's the high level language of math. I think people are just butt-hurt that he used the word 'ultimate'.

dsacco 9y ago
Category theory is not “the high level language of math”. You cannot universally apply category theory to all other fields of mathematics.
- leshow 9y ago
  What fields of math do you think it doesn't apply to?
  - dsacco 9y ago
    Analysis.
    - taktoa 9y ago
      Not true: http://www.math.cornell.edu/~oconnor/sia.pdf http://www.math.cornell.edu/~oconnor/sia.pdf https://ncatlab.org/nlab/show/synthetic+differential+geometry https://ncatlab.org/nlab/show/synthetic+differential+geometr...
      - dsacco 9y ago
        You cannot build all of analysis from category theory. Categories, morphisms and functors are not nuanced enough to encapsulate everything in analysis. The question was not, “Can you restate some results in category theory”, the question was “what cannot category theory build?” What you’re linking to is part of a relatively small corpus outside of e.g. algebraic geometry and topology that has successfully been used for nontrivial results. And as another commenter said, while it can be used for e.g. computation, it is extremely poor for complexity theory. There is a reason why basically any analyst (as opposed to algebraic geometer) will use set theory instead of categories for their work.

proc0 9y ago

Because you cannot discard more detail than what CT has chosen to discard. You start with objects and arrows, and how the arrows compose. Every object has an identity which is another arrow, so in some sense you have just a composition of arrows, and that's it. CT starts really simple, and I don't think there's anything left to hide or abstract.

Koshkin 9y ago
You may be wrong here, there have been some discussions - see for example https://www.quora.com/Whats-more-general-than-category-theory https://www.quora.com/Whats-more-general-than-category-theor...
mbid 9y ago
Every category is an infinity category, but the converse is not true. Thus, infinity categories are more general than categories. But you don't have to go that far: Just remove e.g. identity morphisms from categories, and you get something that is more general than categories. It is of course another matter completely whether these structures are a useful abstraction for what you want to accomplish. I've never come across something reasonable that has arrows and associative composition, but no identities. As for infinity categories, I remain very skeptical that they abstract any problems programmers not concerned with topology (i.e. > 99%) have. So yeah, categories are a reasonable baseline from where you can build. But you can't show anything relevant and non-trivial that holds in any category. You'll need more structure, hence less generality, for that.