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> category theorists tend to turn up, simply name things that other people have spent their lives working on A lot of category theory (at least the more basic

> category theorists tend to turn up, simply name things that other people have spent their lives working on A lot of category theory (at least the more basic parts thereof) is about bringing together concepts that were isolatedly worked on in very different mathematical communities and formulate them in a unified way so that one can see a unifying pattern that was hardly visible before. > What practical problems can I solve with it? If you simply want to work on some topic/problem of your interest, category theory will likely be of limited help (except perhaps for finding a more elegant formulation of the things that you work on). If, on the other hand, you seek deep relationships between topics that to most people look unrelated, getting deeply into category theory is likely not a bad idea. Thus: "from the perspective of someone who is interested in solving practical problem", category theory gives you rather little new problem solving techniques by itself. But category theory gives you a cool lookout to perhaps enable you to apply solution techniques from other areas, where without category theory you would not be able to see the relationship between the technique from a seemingly unrelated area and your problem.

jiggawatts 3y ago
> about bringing together concepts Well, yes, obviously. My analogy is just that: it's like bug collection, bringing together a bunch of samples from many fields into a single museum. My criticism, I think, is still valid. Just because you've renamed things in a bunch of fields with a consistently obtuse verbiage doesn't mean you've understood anything, or improved any of the fields in a material way. There's a lot of promise, but little delivery. Category theory is the string theory of mathematics. There. I said it. I raised a similar critique here before, asking for practical examples from computer science where CT has made a contribution, and someone linked to a CT explanation of automatic differentiation (popularised due to AI). The thing is, AD predated CT substantially, and was well-understood for a long time. The original theoretical description of AD is something I understood, and the CT description of it was impenetrable, and not useful. Category theory is something I've always wanted to succeed. It feels like we're on the cusp of something great, like formalising every part of programming languages and compilers for provably correct transformations and optimisations. That would be amazing, but every time I dig into the promise of CT all I see is some hand-wavy "it's applicable to field X" with literally zero further discussion, examples, or practice.
- synetic 3y ago
  I get the complaint. In the old days people would sometimes say “abstract nonsense” when talking about category theory related topics. It’s increasingly an important part of a basic mathematical education at the graduate level. Eventually there may be a spillover into other areas like computer science where so called practical uses occur. But maybe not. It’s still too early to say. There’s a famous book called Categories for the Working Mathematician. Maybe someday there will be a book called Categories for the Working Programmer. I like that you call it the string theory of math. It’s a more modern way of saying “abstract nonsense”.
  - consilient 3y ago
    > I like that you call it the string theory of math. It’s a more modern way of saying “abstract nonsense”. The sociological analogy is a decent one - they're both ambitious sub-fields of little interest to the average practitioner, with a handful of committed opponents and a wildly overgrown internet hatedom - but their positions in their respective disciplines aren't remotely similar. String theory is a physical theory like any other: pick a spacetime, fill it with dynamical objects, impose some symmetries, and away you go. Category theory, at its most ambitious, wants to be a setting for doing ordinary mathematics in: it belongs in the same general grouping as set theory and higher order logic, not calculus or linear algebra.
  - dkjaudyeqooe 3y ago
    There's this, but the programmer doesn't have to be working: https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ https://bartoszmilewski.com/2014/10/28/category-theory-for-p... https://github.com/hmemcpy/milewski-ctfp-pdf https://github.com/hmemcpy/milewski-ctfp-pdf
- catgary 3y ago
  I mean, the modern language of algebraic geometry and homotopy theory is based on advanced category theory. David Spivak, the author of this book, wrote a great thesis on cobordisms and used categorical techniques to sidestep hard problems in intersection theory. You can also look at some of Ezra Getzler’s work to see how setting up clean categorical abstractions can make it easier to attack hard problems in geometry. In CS, abstractions like monads and effect handlers come from category theory. Plotkin and Power did a lot of super technical work on syntax/semantics adjunctions while sorting out the foundations of effect handlers.
xeonmc 3y ago
So would it be fair to say that category theorists are mostly concerned with categorizing things?
- aleph_minus_one 3y ago
  > So would it be fair to say that category theorists are mostly concerned with categorizing things? The more basic parts of category theory (which even many mathematicians won't be able to surpass) have in my opinion a slight element of this, even though in my opinion there is still more to even this basic level: getting the categorical perspective often gives you elucidations of various areas of mathematics in the sense that it often shows you how some "side definitions" that are studied in some mathematical area (i.e. definitions that are often mentioned in textbooks, but are not central to the study plan of the textbook) turn out to be more important than it seems without having the framework of category theory at hand. But after you have reached this level, you can start formulating deep and far-reaching mathematical conjectures using category theory which would be hard to formulate and appreciate without category theory. Here, category theory goes far beyond categorizing things: new categorical abstractions and conjectures formulated in terms of them often become deep new generalizations of topics studied in long-existing mathematical areas. Just to be clear: this is for mathematics. For computer science, I rather doubt that if the problems that you are working on are not suitable for formulating in terms of category theory, new definitions that would be hard to formulate without category theory are helpful for the computer science topic that you work on. In this sense: in my opinion most currently existing applications of category theory in computer science are indeed about categorizing things.

jiggawatts 3y ago

> about bringing together concepts Well, yes, obviously. My analogy is just that: it's like bug collection, bringing together a bunch of samples from many fields into a single museum. My criticism, I think, is still valid. Just because you've renamed things in a bunch of fields with a consistently obtuse verbiage doesn't mean you've understood anything, or improved any of the fields in a material way. There's a lot of promise, but little delivery. Category theory is the string theory of mathematics. There. I said it. I raised a similar critique here before, asking for practical examples from computer science where CT has made a contribution, and someone linked to a CT explanation of automatic differentiation (popularised due to AI). The thing is, AD predated CT substantially, and was well-understood for a long time. The original theoretical description of AD is something I understood, and the CT description of it was impenetrable, and not useful. Category theory is something I've always wanted to succeed. It feels like we're on the cusp of something great, like formalising every part of programming languages and compilers for provably correct transformations and optimisations. That would be amazing, but every time I dig into the promise of CT all I see is some hand-wavy "it's applicable to field X" with literally zero further discussion, examples, or practice.

synetic 3y ago
I get the complaint. In the old days people would sometimes say “abstract nonsense” when talking about category theory related topics. It’s increasingly an important part of a basic mathematical education at the graduate level. Eventually there may be a spillover into other areas like computer science where so called practical uses occur. But maybe not. It’s still too early to say. There’s a famous book called Categories for the Working Mathematician. Maybe someday there will be a book called Categories for the Working Programmer. I like that you call it the string theory of math. It’s a more modern way of saying “abstract nonsense”.
- consilient 3y ago
  > I like that you call it the string theory of math. It’s a more modern way of saying “abstract nonsense”. The sociological analogy is a decent one - they're both ambitious sub-fields of little interest to the average practitioner, with a handful of committed opponents and a wildly overgrown internet hatedom - but their positions in their respective disciplines aren't remotely similar. String theory is a physical theory like any other: pick a spacetime, fill it with dynamical objects, impose some symmetries, and away you go. Category theory, at its most ambitious, wants to be a setting for doing ordinary mathematics in: it belongs in the same general grouping as set theory and higher order logic, not calculus or linear algebra.
- dkjaudyeqooe 3y ago
  There's this, but the programmer doesn't have to be working: https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ https://bartoszmilewski.com/2014/10/28/category-theory-for-p... https://github.com/hmemcpy/milewski-ctfp-pdf https://github.com/hmemcpy/milewski-ctfp-pdf
catgary 3y ago
I mean, the modern language of algebraic geometry and homotopy theory is based on advanced category theory. David Spivak, the author of this book, wrote a great thesis on cobordisms and used categorical techniques to sidestep hard problems in intersection theory. You can also look at some of Ezra Getzler’s work to see how setting up clean categorical abstractions can make it easier to attack hard problems in geometry. In CS, abstractions like monads and effect handlers come from category theory. Plotkin and Power did a lot of super technical work on syntax/semantics adjunctions while sorting out the foundations of effect handlers.

xeonmc 3y ago

So would it be fair to say that category theorists are mostly concerned with categorizing things?

aleph_minus_one 3y ago
> So would it be fair to say that category theorists are mostly concerned with categorizing things? The more basic parts of category theory (which even many mathematicians won't be able to surpass) have in my opinion a slight element of this, even though in my opinion there is still more to even this basic level: getting the categorical perspective often gives you elucidations of various areas of mathematics in the sense that it often shows you how some "side definitions" that are studied in some mathematical area (i.e. definitions that are often mentioned in textbooks, but are not central to the study plan of the textbook) turn out to be more important than it seems without having the framework of category theory at hand. But after you have reached this level, you can start formulating deep and far-reaching mathematical conjectures using category theory which would be hard to formulate and appreciate without category theory. Here, category theory goes far beyond categorizing things: new categorical abstractions and conjectures formulated in terms of them often become deep new generalizations of topics studied in long-existing mathematical areas. Just to be clear: this is for mathematics. For computer science, I rather doubt that if the problems that you are working on are not suitable for formulating in terms of category theory, new definitions that would be hard to formulate without category theory are helpful for the computer science topic that you work on. In this sense: in my opinion most currently existing applications of category theory in computer science are indeed about categorizing things.