/sci/ - Science & Math

>>10219241
>He might be retarded because he also didn't understand why I broke proving that two sets are equal into 2 parts, proving one is a subset of the other, and then proving the other one is a subset of the first
You just answered your own question

>>10219245
Then my question should be how the fuck these poeple get jobs at universities.

>>10219241
Ask for an advisor who isn't on heroin?

>>10219241
lmao, what the fuck. I'd suggest adding more exposition to explain the notation but it sounds like your advisor just wouldn't give a fuck.

>>10219253
Skills dull over time and publish or perish encourages hyperspecialization. Settle down.

>>10219241
incidentally, what's a good undergrad level book to get into category theory? i've heard good things about "categories for the working mathematician". is this actually intended strictly for graduates/working mathematicians or can an undergrad get through it?

>>10219241
Get a second opinion from a professor at your university. If the new one agrees with the old one, revise what you're doing, if not, switch advisors.

>>10219863
I'm not OP but it kind of depends on your mathematical/comp sci background and what you want to do with category theory (eg. are you only interested in learning category theory for a specific area of math/comp sci or are you interested in learning pure category theory?).

>>10219241
>he doesn't know that to survive in postgrad every proof has to be written as a proof by contradiction
Suppose these two sets are different. Then, one contains something the other doesn't. However, one contains the other, therefore, by contradiction, that's impossible.

>>10219863
There are no "undergrad level" category theory books. 90% of undergrads don't have enough background for it to appear to be anything other than a very, very mild generalization of abstract algebra.

There's a new (compared to Mac Lane) book by Emily Riehl on categories that's really good. It has a fuckton of examples, and I'd suggest you try to read that and if you have no idea what any of the examples are that's a good indication you need to learn more actual math first and then try again.

>>10219908
This one?
https://libgen.pw/item/detail/id/1569697?id=1569697

>>10219912
yes, that one
although it's not necessary to pirate it, it's freely available on her website

>>10219908
This is true but there are early grad books and lecture notes that a strong undegrad student can cover. I'm not the person asking but I took a grad level category theory course in my third year of undergrad and learned a lot. It was pure category theory but used material from all over mathematics and comp sci for the problem sets (so having a background in topology, Galois theory, pure functional programming, (typed) lambda calculus, axiomatic set theory/foundations, few classes in logic, etc... all helped but most of the material could be learned on the fly).

I learned from my own professor's (incomplete) lecture notes and in class material. While I really like them I'm worried that they may be too terse to be used for self study (and I don't want to post them without asking permission). Another student recently asked about Tom Leinster's book and it looks pretty reasonable as well. It's a little less detailed than Reihl's book but it covers all the most important stuff a little sooner.
https://arxiv.org/pdf/1612.09375.pdf

The Oregon Programming Language Summer School (OPLSS) does a quick crash course to category theory most years. It's a little less detailed than the books but it covers the material much faster and focuses the examples on comp sci (functional programming, type theory, and categorical logic). The summer school is aimed at PhDs and Grad students but undegrads can attend with a letter of recommendation and a strong undergrad can understand the material, especially the category theory stuff which is intro level compared to some of the other stuff covered. Here are some good lectures:
https://www.cs.uoregon.edu/research/summerschool/summer16/curriculum.php
You can look up other years here:
https://www.cs.uoregon.edu/research/summerschool/archives.html

>>10219863
That one is kind of hard alone, id suggest it Alongside an introduction to category theory by simons

>>10219241
>be invited to give a talk on fractional statistics
>proving the spin-statistics theorem in 3D as a prelim exercise
>write down the short exact sequence that centrally extends [math]SO(1,3)[/math] to the Spin group
>lift the extension to a [math]G[/math]-bundle setting to get a spin structure via the 3-lemma
>showed that the choice of the representation of the central group [math]\mathbb{Z}_2[/math] by which we extend gives rise to fermion or boson statistics
>one of my colleagues goes "what's the point of this, anon?"
>explain that the machinery can also be used to show the existence of anyons in 2D where the central group is much larger
>"can you use something less abstract?"
>mfw

>>10220463
>he got into a pleb field by mistake

>>10219241
Not everyone studies even the basics of category theory, so the first part's not surprising.
Second part sounds like an argument over style, not about whether something is valid or invalid.

>>10219863
I will also endorse Riehl bc I met her and she's super nice. My favorite by far is Borceux's 3-part handbook. Very readable and has many of the proofs people leave out. For topos theory, Mac Lane and Moerdijk bc you learn category theorem in the form of topos theory, and many of the constructions are quite useful and prose is beautiful. I would put CFTWM last. finally, if you like topos theory, the elephant is fantastic.

>>10219863
Chapter 0 is a great introduction to the categorical world view of algebra

>>10219241
category theory is based af

your prof is a brainlet

>>10220463
anyons are the interface between bosons and bosons and ferms and ferms and bosons and ferms?

what would two orthogonal planes of anyons interacting look like? do they mirror each others movements or are they more 3d than 2d?

>>10220787
Let's work with indistinguishable particles for clarity. In general there are two approaches to understand aynons; the approach I described and the braid group approach; here I'll describe the latter since I have already described the former.
Let [math]M[/math] be a smooth manifold and put [math]Q_N(M) = (M^N\setminus\Delta)/S_N[/math] be the configuration space on [math]M[/math], where [math]\Delta[/math] is the set of diagonals in [math]M^N[/math]. We define the braid group [math]B_N=\pi_1(Q_N(M))[/math] as the fundamental groups of the configuration space.
Now a wavefunction is defined as a section of a (prequantum) Hermitian line bundle [math]H\rightarrow Q_N(M)[/math], on which [math]B_N[/math] acts. To characterize the [math]U(1)[/math]-representations of this action, one can WLOG put particles [math](z_1,\dots,z_N) \in Q_N(M)[/math] on a straight line and draw braid diagrams as they time evolve; the braiding operation gives you the generators [math]\sigma_i[/math] of [math]B_N[/math]. For [math]M = \mathbb{C}[/math] we have the Artin group [math]B_N = \langle \sigma_i\mid \begin{cases}\sigma_i\sigma_j\sigma_i = \sigma_j &; |i-j|<2\\ \sigma_i\sigma_j = \sigma_j\sigma_i &; |i-j|\geq 2\end{cases}\rangle[/math], whose Abelianization [math][B_N]_\text{Ab}\cong U(1)[/math] characterizes the [math]U(1)[/math]-reps. However for [math]M=S^2[/math], we have the condition that [math]\prod_{i=1}^N\sigma_i\prod_{j=0}^{N-1}\sigma_{N-j} = e[/math], hence the [math]U(1)[/math]-reps are labeled by, and hence particles on the sphere can acquire phases of, [math]N[/math]-th roots of unity.
Now notice that we can unbraid any links in [math]M = \mathbb{R}^3[/math], so we have another relation [math]\sigma_i^2 = e[/math] in the braid group. The [math]U(1)[/math]-reps are then labeled by [math]e^{i0} = 1[/math] and [math]e^{i\pi} = -1[/math]. This is why we only get bosons and fermions in 3D.

lol your professor was trying to tell you to remove the fucking stupid category theory from your paper which is on a respectable field like topology
what, are you asking to be the laughingstock of your department?
your professor is giving you a tip, but he doesn't want to discourage you. consider yourself lucky you have people like us to tell you when it's time to hide your... uh... "power" level.

Totally unrelated, just wanted to add some fresh category theory OC. Thank.

>>10220463
>toymaker calls toymakers on toymaking
kek

>>10221693
I want to learn quantales because they came up in some research. How much should I know about frames and locales before making the jump (I've been working through a book on frames and locales after binging a bunch of lecture notes on lattice theory, I have some category theory background but just an intro grad course)?

>>10221771
Fuck, I dunno man.

I've just been reading through Fong and Spivak's Seven Sketches. It's an intro book into category theory, but from a bit of an unorthodox perspective, without any algebra or topology.

Quantales made sense in that context just fine.

>>10220907
It makes me happy to know that turbo autist math geniuses come to the same Japanese image board I do. I've been thinking of getting a math phd myself. I only have undergrad engineering degree and been working in industry a few years. Thoughts?

>>10220907
too much fella. I get sort of what youre saying but it would take me months to get my head around intuitively what those words mean.

>>10219863

>>10221693 and >>10221911 here
I'm an undergrad with no specific math backround, and Seven Sketches has been awesome, specifically alongside John Baez's supplementary course on azimuth.

http://www.azimuthproject.org/azimuth/show/Applied%20Category%20Theory

Highly recommend.

>>10221980
I have a chemE and math degree, and the math you would have done in engineering is nowhere near what you will need if you want to do postgraduate math. Unless it's some numerical math you are interested in, then you might have enough.

>>10221980
Lol good luck. It is well known that 98% of engineers are to retard for even a bachelors in math. Keep dreaming buddy

>>10219863
MacLane's book is a meme. Read Riehl's Category Theory in Context for a general outline of pure category theory. There aren't many prereqs you'll need (an intro to proofs course should suffice), but some of the examples might feel a little unmotivated or go over your head completely. I'd suggest reading Algebra Chapter 0 for your undergrad algebra course first just to see how categorical notions are applied elsewhere in math.

>>10222745
>>10222126
Yeah my math background isn't really that high. I did real and complex analysis and number theory. That's about it for the upper division math major side. I have been doing a lot of machine learning though in work, and not just using TensorFlow or whatever, but actually doing statistical learning.

Do you have any recommendations on places to start? You might be right in that pure mathematics is not the right place for me as it might be too abstract, I am more interested in mathematics in computer science, quantum mechanics, and machine learning/AI.

>>10222813
You'll want to learn linear algebra, measure theory, functional analyses, group/ring/field theory next.

>>10222813
Well i cannot help you with that. If i actually studied something applied I would have to actually take a shower and get a job.

>>10221771
coming from the reverse (not knowing much abt quantales but a decent amount about locales), I can imagine it will be very useful. It would be like learning noncommutative geometry knowing very little of commutative algebra. locale theory is also very fun - peek through the relevant sections of elephant (NOT johnstone's topos theory - it's too terse for me), and borceux's handbook on locales. vicker's topology via logic I think also goes into quantales a bit while reviewing the basics of locales.

>>10221980
>Thoughts?
I have none.
>>10222011
Jacak has a good book on this topic.

>>10223674
but honestly thanks for the explanation!