/sci/ - Science & Math

>>9822919
Because its the most useful thing in math.

>>9822921
>Because its the most useful thing in math.
Since when is category theory in math?

>>9822921
What the heck, you're studying witchcraft

>>9822921
Citation needed. Currently catagory/network theory looks to have the same problematic issues as the varied system theories. And we all know how over-optimistic those were...

>>9822919
What makes a category different from a set?

>>9823540
dude, it's just like type variables in Java code!

>>9823306
Network theory is a meme but without category theory much of modern number theory and virtually all of algebraic geometry just wouldn’t exist.

Because it's very general and sounds like nonsense sometimes.

>>9823555
>without category theory much of modern number theory ... just wouldn’t exist.

>>9823540
Structure. But only locally small categories are enriched in sets, obviously.

>>9823567
>what is arithmetic geometry

>>9823567
Galois Cohomology

>>9823645
arithmetic geometry is just algebraic geometry anyway, and I cut out that part of the quote for a reason

>>9823657
would exist just fine without category theory

>>9823681
But without category theory they'd have to write it in a way other people can understand. How would they keep winning fields medals then?

>>9823681
Arithmetic Geometry does not exist to use number theory to solve geometric problems, it exists to use geometry to solve number theoretic problems.

Saying it is just algebraic geometry is ignorant.

>>9823700
If you somehow think doing homological algebra without category theory would make things easier to understand, you are retarded.

>>9823706
>If you somehow think doing homological algebra without category theory would make things easier to understand, you are retarded.

>>9823706
>Saying it is just algebraic geometry is ignorant.
arithmetic geometry is just algebraic geometry over [math] Spec(\mathbb{Z}) [/math], any other definition would be one that predates category theory

category theory has barely affected the development of modern number theory, it plays no role in analytic number theory, the Langlands program exists without it, most number theory texts don't even mention categories

>>9823742
That looks like you just googled a definition of arithmetic geometry and have no actual experience with it.

>>9823755
>That looks like you just googled a definition of arithmetic geometry and have no actual experience with it.
do you have anything substantial to add?

>>9823755
Probably a good idea to ignore it, friend.

>>9823758
First off, arithmetic geometry is not just over Z. But over Z, finite fields, number fields, rings of integers for number fields, p-adics, etc. Pretty much any base of number theoretic interest.

Also, the usefulness of techniques changes depending over what base your working. Someone working over C is going to be doing very different things from someone working over arithmetic bases.

The whole point of something like etale cohomology is to have a nice cohomology theory for arithmetic geometry, where we constructions like l-adic cohomology and nice results like the etale cohomology of Spec(k) is the Galois cohomology of k. These technqiues are specialized to an arithmetic perspective, because otherwise you are working over C and can use analytic stuff.

Classifying either as “just algebraic geometry” is lazy in the sense the subject changes drastically.

>>9823895
>First off, arithmetic geometry is not just over Z. But over Z, finite fields, number fields, rings of integers for number fields, p-adics, etc. Pretty much any base of number theoretic interest.
I said over Spec(Z), and implicit was being of finite type, which essentially encompasses schemes over finite fields. The rest are not arithmetic geometry

>Also, the usefulness of techniques changes depending over what base your working. Someone working over C is going to be doing very different things from someone working over arithmetic bases.
>The whole point of something like etale cohomology is to have a nice cohomology theory for arithmetic geometry, where we constructions like l-adic cohomology and nice results like the etale cohomology of Spec(k) is the Galois cohomology of k. These technqiues are specialized to an arithmetic perspective, because otherwise you are working over C and can use analytic stuff.
I don't see the relevance of this

>Classifying either as “just algebraic geometry” is lazy in the sense the subject changes drastically.
What else would it be?

>>9823945
>The rest are not arithmetic geometry

Ok I guess people like Schloze aren’t arithmetic geometers.

>I don’t see the relevance of this

It’s like saying the distinction between studying Lie groups and finite groups is irrelevant because they are both just group theory.

>>9822919
>such a fucking meme
Lrn2meme fgt pls

>>9825078
>fgt
Why the homophobia?

>>9824384
>Ok I guess people like Schloze aren’t arithmetic geometers.
But Scholze does work over [math] Spec(\mathbb{Z}) [/math]

>>9822919
Category theory is only useful for doing useless math.

>>9825124
The whole point of perfectoid spaces is to work over fields of mixed characteristic, like the p-adics.

>>9825145
>The whole point of perfectoid spaces is to work over fields of mixed characteristic, like the p-adics.
And so...?

>>9825150
U dumb

This thread embarrasses me.

People who have never worked with any kind of cohomology should not be allowed to discuss category theory. Much like anyone who has never worked with Lebesgue spaces should not be allowed to discuss measure theory. They are the abc stuff, and you are too far down the pleb tier to appreciate what they are for. Imagine a high school kid not knowing what calculus is for. You just sound stupid like that. Keep studying and you'll be able to answer your own question.

>>9825680
You don't need to know any cohomology to appreciate cat theory, basic ring/field theory uses it enough to start understanding why it's useful.

>>9825922
>You don't need to know any cohomology to appreciate cat theory, basic ring/field theory uses it enough to start understanding why it's useful.
You don't need even need ring/field theory, category theory can be appreciated on it's own

>>9825932
No it can’t. Because category theory on its own is pointless.

>>9825945
yes undergraduates love to claim various stupidities on the internet

>>9825945
>Because category theory on its own is pointless.
What do you mean?

>>9825922
>>9825932
You can learn Category theory via Set theory applications. See Lawvere or Spivak's writings.

>>9825945
t. former math student who got his ass kicked once things got a little abstract

>>9826873
The opposite, I was once the undergrad enamored by category theory. When I actually got to the point of needing to learn it, for algebraic topology (and later algebraic geometry), I realized it is just a bunch of definitions and a few near trivial results.

It has no actual depth to it.

you know how like every once in a while you see a proof like the mobius strip solution to the inscribed rectangle problem and worry that you'll never be able to have an insight as profound and elegant as that one? well, category theory is meant to replace that

>>9827156
>cohomology
The results only seem trivial because the structure is so elegant.

>>9827156
>I was once the undergrad enamored by category theory
stopped reading right there

>>9827507
I'm talking about pure category theory. Cohomology is homological algebra and/or topology.

>>9826873
t. LARPer

>>9827514
Because it hit a little too close to home?

>>9828873
>Because it hit a little too close to home?
I've never been "enamored" by category theory.

>>9828897
No one said you were

>>9830212
>No one said you were
No one said anyone said I was.

>>9822919
It isn't a "meme".

>>9823540
Please somebody answer this.

>>9830824
>Please somebody answer this.
Look at definitions of each

>>9830824
Every category is a "member" of itself.
No set is a "member" of itself.

>>9830824
A set is just some elements with some axioms, and you can build on it by defining functions and shit, a category can be passed on set theory, in that case a category is a set of things that satisfy some other axioms, but you don't actualy need sets, it can be classes, or even objects of other categories, a lot of times you just talk of some collection of things without even defining them.

The important point is that the axioms you have for a category are still loose enough so that you can build almost anything on it, just like set theory, but it's also constrained enough so that you can make some very cool theorems that work in any category, or large collections of categories, which abstract a lot of useful things from other areas of math.

If we do category theory over sets then a category is a set A of things we call arrows with the extra rules that
>There exists another set A3 of triplets of these arrows and we call the 3rd element the composition of the first and second one in the triplet
>For each arrow a in A there are two elements r,l, such that (l,a,a) in A3 and (a,r,a) in A3
>If r,l, are present in multiple triplets they will always be in triplets if the above form
I thinks thats all, I'm a bit rusty.

test $G=\Gal(L,K)$

>>9822919
Useless to data science, useless to physics. IOW - useless. Just memeing with stupid dead-end isomorphism diagrams. Abstraction without usefullness. NOT usefull in data structures

>>9830824
If you're talking foundations, then the important part is the set theory is material and category theory is structural. The former gives you faux theorems and the later is closer to logic.

For example, say you model the natural numbers in set theory like so (see e.g. the von Neumann definition and this is standard):
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
0 := {}
1 := {0}
2 := {0,1}
3 := {0,1, 2}
4 := {0,1, 2, 4}
... etc.

Now say you model the ordered pair of sets like so (Kuratowski definition, also the standard):
https://en.wikipedia.org/wiki/Ordered_pair
(x,y) := {{x}, {x,y}}

You can now prove that the number 1 is an element of the pair (0,7).

That's not nice, but an artifact of models.
Such a thing can't happen in category theory, because there, definitions are made via universal morphisms. Those are general forall-there-exists-such-that syntactic expressions of properties, i.e. you define objects by what they do, not by what they are.
E.g. you define natural numbers by how they label stuff and how the successor function interacts with it, and you define ordered pairs by how you can map them to their left and right component.
Because of this, languages of computation are actually in correspondence with cateogries (as well as with proofs)
https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence#Curry%E2%80%93Howard%E2%80%93Lambek_correspondence

4 := {0,1, 2, 3}

PDE best math ever, dude!

>>9830963
This guy knows what's up.

Category theory is crap

>>9830886
I'll look into the correctness of the greentext; either way, that gives me something to think about. You are essentially saying defining a category over a set by interpreting the elements of the set as mappings, and laying out the rules for composition and identity transformations by defining additional sets. This kind of approach makes a lot of sense to me, although it still makes me wonder what classes are...

>>9830963
Interesting. I'm familiar with those set-theoretic models of the natural numbers and ordered-pairs, but I never thought about how they interact...

0 = {}
1 = {0}
(0,7) = { {0}, {0, 7} } = { 1, {0, 7} }

So what you guys are trying to get at is that category theory focuses more on the mappings between things, rather than the things themselves? And this can sidestep issues with sets like the above?

I need to go through the Curry-Howard article. It looks like a good read. For some reason this made me think of Godel.

>>9831472
>Interpreting elements as mappings
Basically. a category is a generilization of a monoid, a one object category with arrows {a,b,c...} is exactly a monoid over the set {a,b,c...}, when you have more than one object in a category you get something with multiple identity elements, where all arrows cant be composed, but when it can be you still have associativity and the identity elements working as expected.

>Classes
I'm sure there's some formal definition, but on category theory it's usualy just taken to be some collection of elements that's too large to be a set.

>>9831472
>although it still makes me wonder what classes are...

Category theorists generally just pretend they are sets that behave how they want them to.

In higher category theory they just straight up model them with sets.

>>9831472
>Focus more on the mappings than the things themselves
Congratulations, lots of people take a long time to get that. It doesn't just focus more on it though, all information is included purely in which mappings can be composed and which mappings are equal. The things themselves carry no structure and are considered black boxes. This allows is to write theorems that work for any thing as long as the mappings have some structure. Take the product for example, it has very modest requirements on the mappings and depending on what the objects are you get a lot of different things, all of which seem different yet follow the same rules
>Category of sets with all functions between them = product is Cartesian product
>Category of real numbers with larger or equal relation = product of two numbers is the maximum of the two numbers
The power of category theory is in defining transformations between categories that preserve these structures even though they look completely different.

>>9830973
I unironically believe this