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Basic Category Theory

arxiv.org

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Re: Basic Category Theory

#11

Post by adamnemecek »

keyboardwriter wrote: post #8
adamnemecek wrote: post #5

Earlier quoted context omitted.

You might be interested in "Algebra: Chapter 0" http://amzn.to/2iYpAn5 "The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics."

Thanks, but is this another textbook that requires knowledge in abstract algebra prior to reading it, or make any sort of previous knowledge assumptions?

You might be interested in this then https://news.ycombinator.com/item?id=13268335

The book doesn't really require much knowledge.

Re: Basic Category Theory

#12

Post by keyboardwriter »

adamnemecek wrote: post #5
keyboardwriter wrote: post #3

Earlier quoted context omitted.

Why do mathematicians use phrases such as "relatively little mathematical background" or "introduction to.." when they assume previous knowledge? I find that really annoying and a turn off from reading most math textbooks that are suppose to be "introductions". Is there an actual good book on category theory for someone that didn't complete a math degree? Additionally, what are applications of category theory to comp…

You might be interested in "Algebra: Chapter 0" http://amzn.to/2iYpAn5 "The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics."

I'll check out this book then. Has great reviews.

Re: Basic Category Theory

#13

Post by cokernel »

keyboardwriter wrote: post #3
petercooper wrote: post #2

"for readers with relatively little mathematical background." A well put together guide, but note that 'relatively little' here means you're OK with some abstract algebra, at least, as the second example of the introduction begins: This example involves rings, which in this book are always taken to have a multiplicative identity, called 1. Similarly, homomorphisms of rings are understood to preserve multiplicative id…

Why do mathematicians use phrases such as "relatively little mathematical background" or "introduction to.." when they assume previous knowledge? I find that really annoying and a turn off from reading most math textbooks that are suppose to be "introductions". Is there an actual good book on category theory for someone that didn't complete a math degree? Additionally, what are applications of category theory to comp…

I can't comment on applications of category theory to computer programming or computer science, but I believe that the book Conceptual Mathematics by Lawvere and Schanuel is a nice introduction to the basic ideas of category theory.

http://www.cambridge.org/catalogue/catalogue.asp?isbn=978052...

I believe this book has been discussed "recently" on HN, but I couldn't find a thread.

Mathematicians write for multiple audiences, but there are two major mathematical audiences for their works: specialists in the same field and researchers in other fields. Phrases like "relatively little mathematical background" normally signal that a work is intended to be accessible to non-specialists, but it's often safe to assume that it is aimed at research mathematicians. I think if a work actually requires relatively little mathematical background, a mathematician is more likely to say something like "no, really, you don't need to know mathematics to understand this!" even when it's not quite true.

Re: Basic Category Theory

#14

Post by qwerty12653 »

keyboardwriter wrote: post #8
adamnemecek wrote: post #5

Earlier quoted context omitted.

You might be interested in "Algebra: Chapter 0" http://amzn.to/2iYpAn5 "The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics."

Thanks, but is this another textbook that requires knowledge in abstract algebra prior to reading it, or make any sort of previous knowledge assumptions?

Having attempted to get through it myself(and having a math degree), I would strongly recommend against Algebra: Chapter 0 as an introductory text on either category theory or abstract algebra.

If you either already have a math degree, or if you're already at the point where you're ready to take an upper-level undergraduate math course, then it may be a decent book for you. Maybe. For example, if you're:

- already very comfortable with linear algebra

- possibly been exposed to a bit of topology in analysis class

- already been exposed to abstract algebra a bit by a more theoretical approach to linear algebra (comfortable with proving things in linear algebra via vector spaces and fields)

Otherwise, I would absolutely steer clear of it. The writing style is also not terribly beginner-friendly, in my opinion.

The closest book I can think of to what you're asking for would be "Conceptual Mathematics: A First Introduction to Categories". Though obviously I don't know whether it's a good fit for your particular background and learning style.

Re: Basic Category Theory

#15

Post by keyboardwriter »

cokernel wrote: post #13
keyboardwriter wrote: post #3

Earlier quoted context omitted.

Why do mathematicians use phrases such as "relatively little mathematical background" or "introduction to.." when they assume previous knowledge? I find that really annoying and a turn off from reading most math textbooks that are suppose to be "introductions". Is there an actual good book on category theory for someone that didn't complete a math degree? Additionally, what are applications of category theory to comp…

I can't comment on applications of category theory to computer programming or computer science, but I believe that the book Conceptual Mathematics by Lawvere and Schanuel is a nice introduction to the basic ideas of category theory. http://www.cambridge.org/catalogue/catalogue.asp?isbn=978052... I believe this book has been discussed "recently" on HN, but I couldn't find a thread. Mathematicians write for multiple au…

Great recommendation. Thank you!

Re: Basic Category Theory

#16

Post by decasteve »

For an even more basic introduction I'd recommend starting with Lawvere and Schanuel's Conceptual Mathematics: A First Introduction to Categories. Requires less abstract algebra. If you have some mathematical intuition then this is the book to start with.

I'm surprised he cites Sets for Mathematics but doesn't include it in his further reading. If you have some university level abstract algebra (groups, rings, vector spaces, topology) then Sets is a quicker start and better reference guide for Category Theory. I started with this book before I had done any abstract algebra and I quickly got out of my depth. But when I did get more algebraic structures under my belt, I had quite a few aha! moments and referred back to this book quite frequently.

I haven't read Awodey's introductory Category Theory book so I can't comment there.

Speaking of Category Theory: Bill Lawvere is an interesting and exceptional 20th century mathematician who probably has less name recognition than he deserves. Lawvere's Wikipedia page: https://en.wikipedia.org/wiki/William_Lawvere

Re: Basic Category Theory

#17

Post by Bahamut »

This article definitely assumes more knowledge than basic abstract algebra - there are also some key facts used from set theory assumed (i.e. In a proof, it is assumed to be known that if a compositiom of two maps is the identity map on one set, and the reverse composition is the identity on the other set, then the map is a bijection - it also assumes knowledge from abstract algebra that the composition of two homomorphisms is a homomorphism).

I have only looked in the first chapter, so I cannot speak further on that atm. That said, I appreciate that a text talks about the universal property - I did not encounter a definition of it in a text when in grad school. I only encountered it because lecturers at my grad program made sure to talk about it.

Re: Basic Category Theory

#18

Post by omginternets »

stephencanon wrote: post #7
keyboardwriter wrote: post #3

Earlier quoted context omitted.

Why do mathematicians use phrases such as "relatively little mathematical background" or "introduction to.." when they assume previous knowledge? I find that really annoying and a turn off from reading most math textbooks that are suppose to be "introductions". Is there an actual good book on category theory for someone that didn't complete a math degree? Additionally, what are applications of category theory to comp…

The mathematical background assumed here is the content of the first couple weeks of the first upper-level course in an undergraduate mathematics degree. It really is "relatively little mathematical background". That said, I second the recommendation of "Algebra: Chapter 0".

Someone (perhaps you, stephencanon?) suggested Algebra: Chapter 0 to me a few weeks ago.

It's fantastic.

Merry Christmas, HN: https://zr9558.files.wordpress.com/2013/11/algebra-chapter-0...

Edit: Happy Kwanzaa too! http://fef.ogu.edu.tr/matbil/eilgaz/kategori.pdf

Re: Basic Category Theory

#19

Post by empath75 »

petercooper wrote: post #2

"for readers with relatively little mathematical background." A well put together guide, but note that 'relatively little' here means you're OK with some abstract algebra, at least, as the second example of the introduction begins: This example involves rings, which in this book are always taken to have a multiplicative identity, called 1. Similarly, homomorphisms of rings are understood to preserve multiplicative id…

I actually laughed out loud when I got to that part. It seemed like the first page and a half he really tried to explain it without jargon and then just gave up.

To be fair, though, who is going to read an introduction to category theory that isn't familiar with abstract algebra?

Someone should write 'an introduction to introductions to category theory'

Re: Basic Category Theory

#20

Post by danharaj »

decasteve wrote: post #16

For an even more basic introduction I'd recommend starting with Lawvere and Schanuel's Conceptual Mathematics: A First Introduction to Categories . Requires less abstract algebra. If you have some mathematical intuition then this is the book to start with. I'm surprised he cites Sets for Mathematics but doesn't include it in his further reading. If you have some university level abstract algebra (groups, rings, vecto…

> For an even more basic introduction I'd recommend starting with Lawvere and Schanuel's Conceptual Mathematics: A First Introduction to Categories. Requires less abstract algebra. If you have some mathematical intuition then this is the book to start with.

It's a great book! Don't forget to do the exercises. It starts off very easy but if you don't do the exercises all of a sudden you'll be lost.

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