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Basic Category Theory

arxiv.org

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Re: Basic Category Theory

#21

Post by empath75 »

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…

I genuinely think you don't need a math background to get a general understanding of category theory, because it's so high level. A lot of it is just drawings, even, not complicated formulae and proofs. It's not category theory itself that's difficult to understand, it's the examples and vocabulary.

I think you really need no more than high school algebra to understand basically what category theory is about.

Re: Basic Category Theory

#22

Post by Koshkin »

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."

> categories, used as a unifying theme in the presentation of the main topics

Categories are not a good way to be introduced to the "main topics" of abstract algebra. Yes, Category Theory can be seen as the "Chapter 0" of a course on abstract algebra, but for beginners a traditional presentation based on sets would be much easier to digest.

There is a similarity between using categories and using functional programming style or a language - as opposed to using an approach to mathematics based on sets and using the imperative programming style or a language: the latter is just more intuitive and therefore easier to learn for a beginner.

Re: Basic Category Theory

#23

Post by GFK_of_xmaspast »

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?

Not sure I see the point in trying to learn category theory without algebra to motivate it.

Re: Basic Category Theory

#24

Post by elberto34 »

I lost interest in math after calculus, when it stopped being about results and more about abstract and complicated squiggles on the page . Elliptic functions are interesting because it seems calculus-like (such as elliptic integrals and the theta series) but this weird set/category theory stuff just doesn't do it for me.

Re: Basic Category Theory

#25

Post by packetslave »

omginternets wrote: post #18
stephencanon wrote: post #7

Earlier quoted context omitted.

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

we're posting copyrighted material to HN now?

Re: Basic Category Theory

#26

Post by c3534l »

empath75 wrote: post #19
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'

Me! I know some abstract algebra, but I'm not going to understand it without some hand-holding. Remind me what a ring is and how and why it relates to categories.

Re: Basic Category Theory

#27

Post by omginternets »

packetslave wrote: post #25
omginternets wrote: post #18

Earlier quoted context omitted.

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

we're posting copyrighted material to HN now?

It's for a good cause.

Re: Basic Category Theory

#28

Post by ky3 »

empath75 wrote: post #19
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'

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

More like the "Prereqs for an Intro to Category Theory."

The author intended his remark that the work is "for readers with relatively little mathematical background" to mean that his readers aren't assumed to have previous exposure to category theory, not that they have zero math.

He's reiterating the Basic in the title "Basic Category Theory."

Category theory is like the refactoring of a lot of math to express commonalities with a view toward recognition and reuse.

The Gang-of-Four OOP patterns book would also seem abstract for anyone sans programming experience.

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