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Category Theory Illustrated – Natural Transformations

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Re: Category Theory Illustrated – Natural Transformations

#41

Post by ctenb »

Twey wrote: post #30
ctenb wrote: post #25

Earlier quoted context omitted.

It's also wrong, since computer science is traditionally mostly about computation, which has nothing to do with CT

Insofar as ‘computation’ is about mapping one state or value to another state or value, it has a lot to do with CT! The question of whether CT is _useful_ for studying computation is different, and there are certainly other lenses you can see computation through that some people would argue are better. But it's hard to deny that they're _related_.

I mean, technically almost all of math can be related to other math one way or another. To say the CT has a lot to do with computation is definitely a stretch. CT is not a recognized Computer Science subject. It's mostly used in the functional programming community to name certain concepts and theorems, but then applied to a specific type system (so it's not actually doing CT, since your restricting yourself to a single category, whereas CT is really about connecting different categories by generalizing over them).

Re: Category Theory Illustrated – Natural Transformations

#42

Post by Iwan-Zotow »

sesm wrote: post #20
siddboots wrote: post #13

Earlier quoted context omitted.

It's just a good set of models to use to think about all sorts of different mathematical systems, kind of like a unified vocabulary. Beyond undergraduate level, category theory these days plays a huge role within many vast fields - e.g., algebraic geometry, algebraic topology, or representation theory.

I think your reply overstates the importance of category theory in mathematics and doesn't give any hint on what it is about. IMO a better reply would be: category theory appeared to unify the concepts around using discrete objects to prove the properties of continous objects in topology, like fundamental groups, homology groups and homothopy groups. It is only practically useful for very advanced proofs like 2nd Wei…

Wasn't that corrected already? I mean categorical definition of Hask?

Re: Category Theory Illustrated – Natural Transformations

#43

Post by coderatlarge »

hamburgererror wrote: post #8

What's the thing with category theory? I see this topic discussed quite frequently here but I don't get it why people are so into it

one of the things i took away about category theory is that it allows you to avoid repeating certain arguments which boil down to so-called “abstract nonsense” ie they have nothing to do with the specific objects you’re dealing with but rather are a consequence of very generic mapping relationships between them. maybe better versed people can give specifics.

as a very broad example there are multiple ways to define a “homology” (ex simplicial, singular, etc) functor associating certain groups to topological spaces as invariants. but the arguments needed to prove properties of the relationships between those groups can be derived from very general properties of the definitions and don’t need to be re-argued from the very fine definitions of each type of homology.

i think.

Re: Category Theory Illustrated – Natural Transformations

#44

Post by griffzhowl »

sesm wrote: post #22
tristramb wrote: post #21

Earlier quoted context omitted.

So category theory is really the theory of composition of mappings. I conjecture that all programming can be seen as just the composition of mappings. If this is correct then category theory is a theory of programming.

You don't need category theory to connect dots with arrows, graph theory is enough for this.

If you allowed infinite graphs maybe. How would you define a functor or natural transformation in graph theory? Seems like you would need to construct a conceptual system that is just equivalent to category theory

Re: Category Theory Illustrated – Natural Transformations

#45

Post by anal_reactor »

I hate this particular mix of prose and formalism. Too complicated to be pop-sci, too informal to be, well, formal. I got to this part:

> We know that two orders are isomorphic if there are two functors, such that going from one to the other and back again leads you to the same object.

And I have no clue what is a functor, nor order. "Functor" wasn't defined, and "order" is defined as "thin category", which in turn remains undefined.

Seems to me like in order to understand this text you already need to understand category theory. If that's the case, then why would you be reading it?

Re: Category Theory Illustrated – Natural Transformations

#46

Post by tristramb »

ctenb wrote: post #23
tristramb wrote: post #17

Earlier quoted context omitted.

Category theory is what you get when you take mappings instead of sets as the primitive objects of your universe. At first this might seem a perverse thing to do as mappings seem more complex than sets, but that is just because traditionally mappings have usually been defined in terms of sets. In set theory you can specify that two sets be equal and you can also specify that one set be an element of another. In categ…

> you can also specify that two mappings compose Two mappings with corresponding domain/codomain have to compose by definition of a category. It's not something you can specify.

Yes. When you are specifying a system you are building the category that you want it to live in.

Re: Category Theory Illustrated – Natural Transformations

#47

Post by shae »

hamburgererror wrote: post #8

What's the thing with category theory? I see this topic discussed quite frequently here but I don't get it why people are so into it

At the 2018(?) ICFP, I sat between John Wiegley and Conal Elliot. They talked about expressing and solving a programming problem in category theory, and then mapping the solution into whatever programming language their employer was using. From what they said, they were having great success producing efficient and effective solutions following this process.

I decided to look for other cases where this process worked.

I found several, but one off the top of my head is high dimensional analysis, where t-SNE was doing okay, and a group decided to start with CT and try to build something better, and produced UMAP, which is much better.

In short, this does work, and you can find much better solutions this way.

(random link https://stats.stackexchange.com/questions/402668/intuitive-e... )

Re: Category Theory Illustrated – Natural Transformations

#49

Post by griffzhowl »

tristramb wrote: post #17
hamburgererror wrote: post #8

What's the thing with category theory? I see this topic discussed quite frequently here but I don't get it why people are so into it

Category theory is what you get when you take mappings instead of sets as the primitive objects of your universe. At first this might seem a perverse thing to do as mappings seem more complex than sets, but that is just because traditionally mappings have usually been defined in terms of sets. In set theory you can specify that two sets be equal and you can also specify that one set be an element of another. In categ…

> Category theory is what you get when you take mappings instead of sets as the primitive objects of your universe.

I'm not sure about that, because you still need some concept of set (or collection or class) to define a category, because you need a set of objects and mappings between them (technically that's a "small" category, but to define any larger category would require at least as much set-theoretical complication).

More exactly, whereas in set theory it's the membership relation between sets and their elements that is basic, in category theory it's the mapping between objects.

Nevertheless, the basic concepts of set theory can also be defined within category theory, so in that sense they're inter-translatable. In each case though, you need some ambient idea of a collection (or class or set) of the basic objects. Tom Leinster has a brilliantly clear and succinct (8 pages) exposition of how this is done here https://arxiv.org/abs/1212.6543

The thing is, even defining first-order logic requires a (potentially infinite) collection of variables and constant terms; and set theory is embedded in first-order logic, so both set theory and category theory are on the same footing in seemingly requiring a prior conception of some kind of potentially infinite "collection". To be honest I'm a bit puzzled as to how that works logically

Re: Category Theory Illustrated – Natural Transformations

#50

Post by rck »

This is fun. But the bit at the beginning about philosophy is not correct. Parmenides did not believe in what we would call essences, but really did believe that nothing ever changes (along with his fellow Eliatic philosopher Zeno, of paradox fame). The idea that change is an illusion is pretty silly, and so Plato and especially Aristotle worked out what's wrong with that and proposed the idea of _forms_ in part to account for the nature of change. Aristotle extended Plato's idea and grounded it in material reality which we observe via the senses, and that's where the concept of essence really comes from - "essence" comes from the Latin "essentia" which was coined to deal with the tricky Greek οὐσία (ousia - "being") that Aristotle uses in his discussions of change.
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