https://www.youtube.com/watch?v=H0Ek86IH-3Y
I recently found this video. Even if it's posted as a joke video, I think threre is still some educational content in it...
I read that the only reason it wasn't adopted we because too much of foundational set theory was already developed, so it simply wasn't convenient. Then, have people attempted to reconstruct mathematics under category theory? Are category theory and set theory logically equivalent or isomorphic or etc.?
A coauthor.
Hello haskellers,
I know this question has been asked in the past, but I'm hoping to get a more modern answer :')
Are there any recommended resources for learning category theory and haskell?
Some background info, I have a math undergrad degree, and I had fun with the pure math, proofs, theorems, etc. so I'd love to dive deeper into the formal math behind category theory.
After graduating, I've had a few different roles before landing a software engineer position about 6 months ago... mainly working with JavaScript/TypeScript/node.js.
I've also worked through the JavaScript Mostly Adequate Guide to FP, which further piqued my interest in FP, also learned some basic Clojure and used it for Advent of Code this year.
Has anyone worked through the MIT course, Programming with Categories? Does this seem like a good place to start? Leafing through the textbook, it seems to strike a good balance between the formal math and introducing haskell basics. I was thinking of working through it, then work through a more in-depth haskell resource after (i.e. learn you a haskell or something).
What did you use to learn category theory and/or haskell?
Thanks in advance for your input!
I dived into category theory recently(reading Basic Category Theory by Leisner), although I don't have a strong background in abstract algebra. I learned some basic group theory on my own and I only know the definitions of the rest of important structures. I understand some of the examples from the book on categories, mostly about groups/monoids, but I think I don't really understand the ones involving more complicated structures well enough. My question is whether my approach is fine considering that I learn CT just out of interest, or I will miss really a lot without further knowledge in algebra? If I need algebra, is there any "crash course", containing just enough information for understanding examples from CT, missing some details which are usually covered in standard algebra courses?
Im currently wrapping up my math degree and in my Number Theory class my prof briefly brought up category theory. Seemed insanely interesting but insanely complex. Id love to learn more about it but my university doesnt offer any classes for it.
My question is - are there any free textbooks/videos/etc. that I can use to learn category theory in my free time? Im pretty literate in complicated math, so feel free to give any recommendations you got :-)
*examples
This may be a silly question, but why are natural transformations defined in such a way that both functors are linked to the same categories?
That is, is there some reason why it wasn't defined something like this:
If you have two functors F: A -> B and G: C -> D, then a natural transformation N:
F -> G would consist of a morphism from F to G, and two functors Nx: A -> C and Ny: B -> D,
such that obvious diagram with F, G, Nx, and Ny commutes.
Is there a reason for restricting both F and G to the same pair of categories, and is there a name for the generalized version above, or is the above an entirely uninteresting generalization, or...?
Thanks
I'm interested in studying just enough Category Theory to feel comfortable using it in other areas like Representation Theory, PL-theory and so on.
I'm hoping to follow Conceptual Mathematics chapter by chapter because it looked rather casual. I would also like to look into some other important topics like the Yoneda Lemma. If there's another book others prefer, I'm open to suggestions.
I'm also up for diving into specific areas that use category theory, though unfortunately I don't know a thing about Alg. Topology or Alg. Geometry. I do know Haskell and some Type Theory though if that helps.
Ideally this would be on discord. If you're shy about talking, we can use it to just post questions to eachother.
Please leave a comment outlining why you chose your answer. (The correct answer will be posted in the comment section once the poll is over)
I'm working through Milewski's excellent intro course to Category Theory, and just learned about the connection between functions and algebraic exponentials. Functions can be written as b^a, where a is the type of the input, and b the type of an output. The size of the space of morphisms between a and b is b^a, where b and a here sort of stand in for the cardinality of each type (not sure if I can talk about cardinality if a or b are not sets).
I'm wondering if the size of b^a has been tried out as a complexity term in a model-fitting context, and can tell you anything about model complexity that the number of free parameters in a model cannot.
Thanks!
I was pleasantly surprised this morning to find a new video from Professor Borcherds on youtube. For those not in the know, Richard Borcherds is a UC Berkeley professor and Fields Medalist known for his work in the notoriously strange and wondrous area called Monstrous Moonshine. His PhD advisor was none other than John Conway. Last year he started a youtube channel, https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ/videos in which he delivers high level exposition of various subjects. He suddenly stopped posting in the beginning of May, but happily it seems he's fine and was just taking a break, as he's now back with the first video in a series on category theory. Rejoice!
https://www.youtube.com/watch?v=JOp7mH72Jlg&ab_channel=RichardE.BORCHERDS
Can someone help me with this? I don't understand what the error message means:
Category.idr:
public export
record Category where
constructor MkCategory
object : Type
morphism : object -> object -> Type
identity : (a : object) -> morphism a a
compose : {a, b, c : object}
-> (f : morphism a b)
-> (g : morphism b c)
-> morphism a c
leftIdentity : {a, b : object}
-> (f : morphism a b)
-> compose (identity a) f = f
rightIdentity : {a, b : object}
-> (f : morphism a b)
-> compose f (identity b) = f
associativity : {a, b, c, d : object}
->(f : morphism a b)
->(g : morphism b c)
->(h : morphism c d)
->compose f (compose g h) = compose (compose f g) h
Functor.idr:
import Category
record CFunctor (cat1: Category) (cat2: Category) where
constructor MkFunctor
mapObj : object cat1 -> object cat2
mapMor : {a, b : object cat1} -> morphism cat1 a b -> morphism cat2 (mapObj a) (mapObj b)
preserveId : {a : object cat1} -> mapMor (identity cat1 a) = identity cat2 (mapObj a)
preserveCompose : {a, b, c : object cat1}
-> (f : morphism cat1 a b)
-> (g : morphism cat1 b c)
-> mapMor (compose cat1 f g) = compose cat2 (mapMor f) (mapMor g)
Output:
$ idris2 Functor.idr
Main>Welcome to Idris 2. Enjoy yourself!
1/2: Building Category (Category.idr)
2/2: Building Functor (Functor.idr)
Error: While processing
constructor MkFunctor. Can't bind implicit Main.{c:546} of type (Main.Category.(.object) cat1[9])
)