/sci/ - Science & Math

Learn Discrete Math with basic proofs and take a course on Abstract Algebra with another course that is focused on Group Theory and you will understand that first sentence.

Also, by that time you will have said "maturity"

>>9651992
What book is this so I can avoid it? This seems to be a common problem among “beginner” category theory related text. “Conceptional mathematics” is suppose to be a beginner category theory text and it starts out deceptively easy with basic set theory but at some point in the book it feels like the swimming pool drops from 2ft of water to 100ft without much warning.

https://arxiv.org/pdf/1803.05316.pdf Is suppose to be an introductory book to category theor. I haven’t read it yet, but I don’t have high hopes for it based on my experience with previous “beginner” books.

>>9652024
>What book is this so I can avoid it?
Emily Riehl - Category Theory in Context

>>9652030
Thanks. I won’t get it. I’ll probably go with what >>9652008 recommended. Feel free to check out the book I linked (it wouldn’t surprise me if it suffers from the same problem).

>>9651992
Z_6, the group of six elements mod 6, e.g. with addition
0+2=2 mod 6
1+2=3 mod 6
2+4=0 mod 6
3+5=2 mod 6
3+3=0 mod 6
etc.
can also be recognized as a subgroup of Z_12. To this end, double all the values and do the same computations
E.g. the above equalities
0+4=4 mod 12
2+4=6 mod 12
4+8=0 mod 12
6+10=4 mod 12
6+6=0 mod 12.

The groups are abelian, as e.g. 2+4=4+2.
The doubling map is not surjective, as e.g. you can't get 5 (in Z_12) by doubling anything (from Z_6). A subgroup is one that is part of a bigger one, akin to the example above. A normal subgroup is one where if you shuffle the elements around in a certain way, nothing changes. All subgroups of abelian subgroups like Z_n are normal anyway, btw. A quotient group is when you take a big group and create a smaller group out of them by creating objects that are actually bundles of the previous ones. Just for examples, you may consider the pairs of natural numbers (x,y) and you may consider the rational numbers x/y, that are also "pairs of numbers", except numbers like 14/6 and 7/3 and 21/18 are actually the same. So you can consider the rational numbers as sets of pairs of numbers, where in one of those sets are the pairs such as "7/3:={(14,6), (7,3),...}." This is not an example of a quotient group, but such quotient constructions are common.

So much for some of the words used in
>A group extension of an abelian group H by an abelian group G consists of a group E together with an inclusion of G → E as a normal subgroup and a surjective homomorphism E ↠ H that displays H as the quotient group E/G.

cont.


You see a lot of those concepts can be exemplified with simple objects. They are only scare as long as you have no connection to them in that sense.
Of course if you only talk in generalities, you'd have to remember a lot. Maturity is being able to look at stuff you know and being able to recognize predicates like the ones discussed in the book in things you know. And, better yet, actively scan your mind and old notes for where those definitions apply.
Algebraic geometry in particular - one of the biggest takers of category theory, next to algebraic topology - was developed in this language. So there, actually, you sometimes set up objects abstractly without known the particulars in great details. That's what makes it abstract but it also has a charm.

Here is a SE post for CS btw., maybe it helps.
https://cs.stackexchange.com/questions/3028/is-category-theory-useful-for-learning-functional-programming?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa

I meant
14/6 and 7/3 and 21/9

>>9652088
how does category theory apply to linear algebra?

>>9651992
one actually studies category theory only after one already knows most of category theory through examples.

>>9652111
That's a bit like asking
>how does language apply to a theather play
I talked about the group example also totally in terms of normal undergrad abstract algebra. Category theory on its own is a language and, as a theory, has few theorems.
If you break up the initial category theory axioms from the 40's to richer structures, you get more of what looks like a mathematical theory, but then it either starts to mirror set theory or homotopy theory.
To answer your question, though, category theory puts focus on some small but rich concepts that were always there but drops everything around it. Universal properties and arrows in particular. If you e.g. want to talk about the tensor product in terms of what it does to you, arrows and diagrams are the natural way to talk about it and that certainly is linear algebra related.
I can post some introductory pages from a book that I really like, if you promise you read it. It gives what I want to all an almost emotional preface to stating the axioms of a category.

>>9652166
>I can post some introductory pages from a book that I really like, if you promise you read it. It gives what I want to all an almost emotional preface to stating the axioms of a category.
what book?

>>9652181
Abelian Categories

>>9651992
>muscular
>dyke haircut
Take me.

>>9652210
she's the author btw

>>9652212
Yeah, google image search told me that. I want to cum in her.

>>9652222
I don't want you to cum in her.

>>9652280
kek

>>9652088
great explanation, thanks

>>9652036
>>9652030
>>9652024
>>9651992
It's a great book actually. It has tons of examples of all flavours. You don't really need any background - if you had actually read to completion the first chapter, it gives people that have had a taste of mathematics (what is the point in learning cat theory otherwise desu) a chance to see the power and usefulness that the unifying approach of cat theory has. The tons of examples can be skipped of course, and you may lose motivation (cat theory was inspired and developed from ideas of algebraic topology after all), but they are littered throughout so that anyone that has had some experience can take something from it.