/sci/ - Science & Math

its literally the comfiest

>>8378830
spotted the analyst

category theory is pure philosophy, not mathematics

Idk when brainlets will realize how fucking easy category theory makes things that could be a lot harder. It's so fucking usefull

topology>algebra>analysis

fight me

>>8379037
Why would I fight you when you're right?

>>8379037
How about algebraic topology / homology / cohomology?

>>8379198
>How about algebraic topology / algebraic topology / algebraic topology
Rofl please end yourself.

>>8379208
I'm asking you to r8 them you retarded faggot

>>8379210
>how do you rate one thing
You're one dense motherfucking dumbcunt aren't you lmao

>>8379210
you obviously know nothing about any of them if you think those 3 can somehow be ranked

please leave this board and log out of your account forever

No-one does category theory for the sake of category theory (except for maybe autistic computer "scientists"). It's like claiming english is an area of math because people write proofs in it. You only use category theory as framework to explain other things like algebraic topology or geometry.

>>8379235
http://arxiv.org/list/math.CT/recent

>>8379241

Did you even look at those submissions? Pretty much all of them involve some other topic like operator algebras, algebraic topology, or quantum physics.

>>8379037
Algebra and topology are dual to each other.

Analysis will eventually be assimilated into them using category theory.

>>8379242
the language of category is the natural language for algebra

>>8379208
>implying there is a partial ordering on the mathematical topics

>>8379380
I know category theory but not much algebra and am struggling to formalize a lot of basic algebra concepts in category theory. For instance the notion of a group action, what category does it live in? Is it a functor? What the hell is it actually from the perspective of category theory??

It seems there are an excessive number of resources for introducing category theory to people with an algebra background but I can't find a single book that teaches algebra to people with a category theory background. All this talk about category theory being a viable foundation of mathematics and no books can I find that actually use it that way.

number theory is the thinking man's area

prove me wrong

>not real and complex analysis
Brainlets

>>8379480
if you care at all about real analysis it's because you haven't been exposed to any of the far more beautiful mathematics

>>8379210
A ranking could be made insine alg top, and I'd rank cohomology better than homology because of the richer structure. One could naturally argue that homology is better refering to certain restrictions in cohomology making sure things don't fall apart, such as the assumption that a space is of finite type for the Künneth formula to work.

>>8379403
>For instance the notion of a group action, what category does it live in? Is it a functor?
In many (and maybe all) cases, a group [math]G[/math] acting on an object [math]X[/math] in a category can be considered by focusing on the underlying object [math]|G|[/math] of the group and a morphism [math]|G| \times X \to X[/math], where [math]|G|[/math] is an object in the category at hand. In the category of groups (and homomorphisms), it is just a complicated way to state the definition of a group acting on a group.

To make things more concrete, two examples shall be given. In the category of topological spaces (and continuous maps), equip the underlying set of any group with the discrete topology to make the operation and inversion continuous and the underlying space [math]T_1[/math], so that your choice of a group becomes a topological group.

Analysis is pig disgusting
Algebra is fun

>>8379210
gr8 b8 m8

i r8 8

>implying anime grills know anything about math
next time post meganeko qts or velma

>>8379700
Pretty much this. We like cohomology more than homology for the same reason we like Postnikov towers better than Moore towers: the latter is derived from the more fundamental former, and that derivation loses vital structure. (Moore towers are homotopy fibers of Postnikov towers.)

Also, Animenon, I made a new post for you to read. The image is a visualization of the second Hopf fibration.

>>8378847
It's nice to see smile

Everything begins and ends with Functors.

>>8378830
It's actually number theory, but thank you for asking.

>>8379847
>ends with Functors
pun intended?

>>8379797
neh

>>8379808
Yes, I read it. I'm working on, but I have no experience in K-theory and I'm actually working atm, so it may take some time to give an answer I'll be satisfied with. I hope I can atleast give some ideas for you, though.

>>8379889
No rush. I just wanted to keep you posted on my progress with handling spectral sequences morally.

K-theory in full generality is rather straightforward; it's basically the distillation of spacial relationships (fibre sequences) into algebraic equations (inside of an abelian group). I'm still learning a lot of it right now, but that seems to be the general spirit of it.

>>8379862
kek
>everything ends and coends with functors

>>8379910
I'll check that stuff out. Your hunch may be correct, as simple ideas leading to rich theory are common. Nice of you to keep me informed!

>>8379483
>what is functional analysis
>what is riemannian geometry
>what is number theory
If you think you can do serious math with no analysis, you're seriously mistaken

>>8379403
Just read MacLane's "Categories for the working mathematician", it's easily available on the net

>>8379037
>topology with algebra
That's like pizza without pineapple

>>8380105
>problems involving specific structures instead of generalizations
>real math

>>8379797
They do this shit in high school there lad.

>>8379403
A group is a groupoid with one object, and a group action is a functor from a group to another category.

Or, given a category where internal groups can be defined, you can define an internal action as a morphism [math]G \times X \to X[/math] satisfying certain axioms, where G is the internal group.

>>8380135
literally kill you'reself if you put pineapple on pizza

>>8380357
>implying it isn't mouthwateringly delicious

>>8380424
it's not good on or off pizza you heretic

>>8380105
Functional analysis and really all sorts of geometric spaces enjoy the generality of higher category theory. Statements in harmonic analysis, for example, are very simple in the language of category theory.

>>8380828
I don't think so to be quite honest. Especially talking about proofs and not only statements. Can you show an example?

>>8380861
I was thinking of the fact that taking group characteristics lets you dualize between discrete groups and compact groups. That's really at the heart of abstract harmonic analysis.

>>8379473

number theory is autism

>>8379403
Look into monoidal categories.

>>8379403
Alternately, a group action is just a functor from the group.

>>8381248
>>I was thinking of the fact that taking group characteristics lets you dualize between discrete groups and compact groups.
that sounds like stone duality and all its generaliztions

>>8381491
They are both special cases of dualizing objects and Isbell duality, yes. A dualizing object is one such that mapping into it yields an adjoint equivalence with the opposite category, and then maps into that object's opposite brings you back.

can someone recommend a book on category theory suitable for a pleb who only knows calc + linear algebra?

>>8380828
It's not really going against my point. Also, you can formulate things any way you want, at some point you are going to have to get your hands dirty if you do want to prove anything. Complexity doesn't disappear.

>>8383892
Complexity is moved onto the language, but I see your point. This is a challenge of mine in a current collaboration.

>>8383363
I'd say you should probably look into group theory first. Calc gets you nowhere but maybe through some examples, and linear algebra doesn't focus on the things you should be focusing on. Just a little bit of group theory to get you started, as the reasoning in the very beginning of category theory is very much similar. Basically, just get yourself comfortable with symmetry groups and maybe kernels of homomorphisms, and you'll already understand a lot more than with what you know now.

But, the more you know the better, as the examples are really various. Atleast in Awodey's book.

>>8383363

>>8384021
Lawvere is so fucking based.

I like Mathematical physics with the focus on stastics.

>>8384076
>stastics
That's not a thing you stupid idiot.
>statistics
>in (mathematical) physics
Goddamn retard.

>>8384109
>what is statistical physics?

>>8384125
>implying stat mech is (mathematical) physics
Holy shit you can't be this fucking stupid.
>literally can't read

>>8384021
thanks pham

>>8379473
>slut
what did terry mean by this

>>8385388
kekarino

I can't find it now, but I remember reading a paper where the author defined a function called Cluster Utility (or CU for short) to measure the validity of a clustering. In his definition, he evaluated this function on the kth cluster, and wrote it like this:

[math] CU(C_k) [/math]

>>8378830
I like categories.

What's up.

>>8378830

>>8383363

No, stick with the calculus-analysis track, learn some neat functional analysis or mathematical physics, that stuff is not only more interesting, but it is more applicable too.

>>8378830
[math]Category X Graph THy. X Spectral Thy.[/math]

>>8378860
can be applied with*

>>8385467
You started a category theory bash thread and nobody seems to be on your side.

>>8386312
>>8386312

I didn't start this thread. But I certainly understand where the op is coming from. Why immerse onesself this kind of algebraic jerkoff when analysis lends itself immediately to both pure and practical considerations. Analysis is unfairly bashed by kiddies fresh out of their intro to groups courses and insecure autists on sci.

>>8386408
To be fair, I don't think analysis should be bashed, as it is definitely a vibrant field with some really elegant arguments. I was shitposting. With that said, category theory and algebra are just as useful. It depends on where you are trying to apply the fields. Want to produce physical calculations relating to physical things, or really relating to anything with continuous variation? Analysis all the way. Want to find the thread connecting different mathematical constructions for easier and more fruitful abstraction and extension? Category theory is the field for you.

They really aren't comparable, but category theory is clung to by lots of mathematicians because it exemplifies the age-old idea of mathematics being incredibly abstract and out of reach. Category theory is to mathematics as philosophy is to human cognition. Analysis and category theory really can't be compared.

>>8386433
admittedly, I will use commutative diagrams in differential geometry, but like a well aged curmudgeon, I find much of the abstraction in twentieth century mathematics to be a touch distressing. I have always felt that any mathematician should atleast feign half concern for practical matters and that obsessions with increasing levels of abstraction divert inquiry away from these areas and into useless directions. Clearly in this century you're not going to find too many Euler-types who work in number theory and celestial mechanics concurrently. However, I still hold this to be an ideal; mathematics is and ought to be the most versatile field, and analysis is, in my mind and without question, its most versatile sub-discipline.

so that's my sort of "analysts apology".

anybody got some interesting documentaries I could watch tonight?

>>8386477
Yes, that's a fair view to hold. I simply don't reflect your sentiments; mathematicians and mathematics do not form coherent wholes. We are people who are merely linked by the level of rigour in our arguments. When you think of it that way, it makes a lot of sense for everything else to vary wildly. I have an overall philosophical and abstract mind, and application just turns me off. That does not mean every mathematician ought to share my view, it just means I am a mathematician with a personal preference. I think a lot of issues arise from the fact that there are relatively few mathematicians, and we have always had to sort of band together since so few people can really appreciate what we study. So, there is a cohesive force which is fighting against the fact that we all hold our own philosophies on what mathematics "should" be.

>>8386486

turned off by applications?

I thought I wouldn't like numerical analysis, that was before I did any. I've found that there is very little in the subject which I've found truly turns me off, I can even amuse myself with problems from the dummit and foote book, or maybe lang, but that's as much algebra as I'd ever do.

do you teach? are you one of those guys who hates teaching introductory calculus?

>>8386520
I am a sort of strange case: a researching undergrad with a few professors and grad students working to get me into the graduate program without a bachelor's degree. I am actually doing some applied collaborative work, but my role is to help put previous results on a firm categorical basis.

I actually would enjoy teaching calculus, because it's an opportunity to introduce people to what mathematical thinking is really about. My ambivalence for the subfield would not get in the way, I don't think.