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Man everyone seems to be talking about category theory lately and I still have no idea what it is except some general sense in which different types of mathemat

Man everyone seems to be talking about category theory lately and I still have no idea what it is except some general sense in which different types of mathematical objects can operate on each other (which is probably wrong). Looks like I should probably actually put the effort in and learn about it.

themonoid 6y ago
Study category theory if you would like to understand "monads are just monoids in the category of endofunctors." Study the flatMap function if you would like to apply category theory to life. A monad is just an object with a sensible flatMap method.
- Fishysoup 6y ago
  Normally I'd go with the second, but I'm trying to learn physics as a hobby and, you know, technologically save the world with my messiah complex apparently, and I want to keep tabs on things that may be insightful to know about
  - themonoid 6y ago
    Just in case you are serious about having a messiah complex, I am prone to having a messiah complex, thanks to schizoaffective disorder. It took me a while to figure out which medications actually help me, without too many unpleasant side effects, but I am very grateful to have discovered the cocktail that works for me now.
    - Fishysoup 6y ago
      i'm glad to hear you've found something that helps you. I was joking about the messiah complex, though it is something my therapist has made the occasional joke about too when i get over-ambitious about things i wanna do (he did specify i don't seem to actually have it, though - just good old-fashioned depression).
      - anonytrary 6y ago
        > just good old-fashioned depression So you're normal (half joking). Humor is a good medication for depression, and at least you can joke about it.
mncharity 6y ago
Just a cautionary note - category theory has seemingly been at "There's tremendous potential here! We just need to find a big win to make that clear to everyone else..." for quite a few years now.
- minkowski 6y ago
  This is rather uncharitable. Category theory is an essential part of the vocabulary of 20th-century mathematics. Large swaths of algebra, topology, geometry, and logic are fairly inextricably formulated in this language. Similarly, it seems irreplaceable for certain parts of programming language theory (arguably due to its connection to mathematical logic). There's certainly a community trying to bring a category-theoretic approach into other fields such as statistics, economics, or other areas of computer science, but it's too early to say it's been "quite a few years".
  - T-A 6y ago
    I think parent meant big wins in physics, where the search started well over a decade ago. See e.g. the dates of the references here: https://en.wikipedia.org/wiki/Categorical_quantum_mechanics#References https://en.wikipedia.org/wiki/Categorical_quantum_mechanics#...
    - GoblinSlayer 6y ago
      QM is just linear algebra, no need to turn it into Haskell.
- Kednicma 6y ago
  I'm obliged, I suppose, to list off a few big wins. From the 50s and 60s, we have classic theorems which use abstract nonsense to generalize big statements about entire classes of objects, like Freyd's adjoint functor theorem [0], Yoneda's lemma [1], and Lawvere's fixed-point theorem [2]. Yoneda's lemma is the slogan that "an object is equivalent to the arrows coming/leaving it", but formal. Lawvere's theorem is a deep and permanent generalization of Gödel, Tarski, Turing, et al. on incompleteness and undecideability. Starting in the 60s and continuing to the present, there's been a theme of exploring category theory as a prime foundation for maths. There's been complete foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular. On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations. [0] https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_functors https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_fu... [1] https://en.wikipedia.org/wiki/Yoneda_lemma https://en.wikipedia.org/wiki/Yoneda_lemma [2] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf [3] https://ncatlab.org/nlab/show/ETCC https://ncatlab.org/nlab/show/ETCC [4] https://arxiv.org/abs/1212.6543 https://arxiv.org/abs/1212.6543
Kednicma 6y ago
As grand-nibling comment suggests, it's primarily about vocabulary and definitions, but also how objects can be put together to form new objects. In combination, we end up with "universal properties", behaviors of objects which are the same in many different contexts. If you want to get some surface-level mind-blown experiences, then there are some cool pages out there, in addition to the papers I already linked, which have great diagrams. [0] shows a periodic table which relates categories to truth values and sets, which is helpful for understanding what folks mean by "category theory is just as good as set theory for foundational work". More mind-blowingly, category theory is "formally formal", which means that we can use category theory to formalize its own concepts. This leads to n-category theory. Because of this, we have tables like [1] which relate category theory's own building blocks to type theory and logic. Category theory was originally the study of natural transformations. Any time that you find enough data to define a natural transformation, then that's interesting; the component categories are then also interesting objects of study. That's still what motivates its presence in physics. There are at least two other perspectives though, a logical perspective (a category is a deductive system for a formal logic) and a spatial perspective (a category is a topological space), and they're both rich as well. But the richest perspective is going to come after you've explored all three. [0] https://ncatlab.org/nlab/show/periodic+table https://ncatlab.org/nlab/show/periodic+table [1] https://ncatlab.org/nlab/show/computational+trinitarianism https://ncatlab.org/nlab/show/computational+trinitarianism
mathgenius 6y ago
It helps if you get interested before putting the effort in.... Check out this breezy youtube from superstar Tai-Danae Bradley: https://www.youtube.com/watch?v=wiadG3ywJIs https://www.youtube.com/watch?v=wiadG3ywJIs

themonoid 6y ago

Study category theory if you would like to understand "monads are just monoids in the category of endofunctors." Study the flatMap function if you would like to apply category theory to life. A monad is just an object with a sensible flatMap method.

Fishysoup 6y ago
Normally I'd go with the second, but I'm trying to learn physics as a hobby and, you know, technologically save the world with my messiah complex apparently, and I want to keep tabs on things that may be insightful to know about
- themonoid 6y ago
  Just in case you are serious about having a messiah complex, I am prone to having a messiah complex, thanks to schizoaffective disorder. It took me a while to figure out which medications actually help me, without too many unpleasant side effects, but I am very grateful to have discovered the cocktail that works for me now.
  - Fishysoup 6y ago
    i'm glad to hear you've found something that helps you. I was joking about the messiah complex, though it is something my therapist has made the occasional joke about too when i get over-ambitious about things i wanna do (he did specify i don't seem to actually have it, though - just good old-fashioned depression).
    - anonytrary 6y ago
      > just good old-fashioned depression So you're normal (half joking). Humor is a good medication for depression, and at least you can joke about it.

mncharity 6y ago

Just a cautionary note - category theory has seemingly been at "There's tremendous potential here! We just need to find a big win to make that clear to everyone else..." for quite a few years now.

minkowski 6y ago
This is rather uncharitable. Category theory is an essential part of the vocabulary of 20th-century mathematics. Large swaths of algebra, topology, geometry, and logic are fairly inextricably formulated in this language. Similarly, it seems irreplaceable for certain parts of programming language theory (arguably due to its connection to mathematical logic). There's certainly a community trying to bring a category-theoretic approach into other fields such as statistics, economics, or other areas of computer science, but it's too early to say it's been "quite a few years".
- T-A 6y ago
  I think parent meant big wins in physics, where the search started well over a decade ago. See e.g. the dates of the references here: https://en.wikipedia.org/wiki/Categorical_quantum_mechanics#References https://en.wikipedia.org/wiki/Categorical_quantum_mechanics#...
  - GoblinSlayer 6y ago
    QM is just linear algebra, no need to turn it into Haskell.
Kednicma 6y ago
I'm obliged, I suppose, to list off a few big wins. From the 50s and 60s, we have classic theorems which use abstract nonsense to generalize big statements about entire classes of objects, like Freyd's adjoint functor theorem [0], Yoneda's lemma [1], and Lawvere's fixed-point theorem [2]. Yoneda's lemma is the slogan that "an object is equivalent to the arrows coming/leaving it", but formal. Lawvere's theorem is a deep and permanent generalization of Gödel, Tarski, Turing, et al. on incompleteness and undecideability. Starting in the 60s and continuing to the present, there's been a theme of exploring category theory as a prime foundation for maths. There's been complete foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular. On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations. [0] https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_functors https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_fu... [1] https://en.wikipedia.org/wiki/Yoneda_lemma https://en.wikipedia.org/wiki/Yoneda_lemma [2] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf [3] https://ncatlab.org/nlab/show/ETCC https://ncatlab.org/nlab/show/ETCC [4] https://arxiv.org/abs/1212.6543 https://arxiv.org/abs/1212.6543

Kednicma 6y ago

As grand-nibling comment suggests, it's primarily about vocabulary and definitions, but also how objects can be put together to form new objects. In combination, we end up with "universal properties", behaviors of objects which are the same in many different contexts. If you want to get some surface-level mind-blown experiences, then there are some cool pages out there, in addition to the papers I already linked, which have great diagrams. [0] shows a periodic table which relates categories to truth values and sets, which is helpful for understanding what folks mean by "category theory is just as good as set theory for foundational work". More mind-blowingly, category theory is "formally formal", which means that we can use category theory to formalize its own concepts. This leads to n-category theory. Because of this, we have tables like [1] which relate category theory's own building blocks to type theory and logic. Category theory was originally the study of natural transformations. Any time that you find enough data to define a natural transformation, then that's interesting; the component categories are then also interesting objects of study. That's still what motivates its presence in physics. There are at least two other perspectives though, a logical perspective (a category is a deductive system for a formal logic) and a spatial perspective (a category is a topological space), and they're both rich as well. But the richest perspective is going to come after you've explored all three. [0] https://ncatlab.org/nlab/show/periodic+table https://ncatlab.org/nlab/show/periodic+table [1] https://ncatlab.org/nlab/show/computational+trinitarianism https://ncatlab.org/nlab/show/computational+trinitarianism

mathgenius 6y ago

It helps if you get interested before putting the effort in.... Check out this breezy youtube from superstar Tai-Danae Bradley: https://www.youtube.com/watch?v=wiadG3ywJIs https://www.youtube.com/watch?v=wiadG3ywJIs