Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Basic Category Theory

arxiv.org

Post reply on HN
Print view View original

41–50 of 92 posts

Re: Basic Category Theory

#41

Post by j2kun »

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?

I think the first chapter definitely has no prerequisites (it introduces sets and categories), but it's also a core mathematics textbook intended for upper-level undergraduate and early graduate students in pure mathematics. So it re-teaches all of abstract algebra, and quite a bit beyond, using category theory as a unifying principle. It's a book intended to give you a mature perspective and prime you for research.

Re: Basic Category Theory

#42

Post by alexott »

mathgenius wrote: post #29

Tom Leinster wrote some things about open sourcing the book here: https://golem.ph.utexas.edu/category/2017/01/basic_category_...

There "Category Theory in Context" book by Emily Riehl is also freely available at https://golem.ph.utexas.edu/category/2016/11/category_theory... (I didn't read it yet)...

Re: Basic Category Theory

#43

Post by [deleted] »

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…

[deleted]

Re: Basic Category Theory

#44

Post by jinfiesto »

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 don't understand why this is down-voted. This is probably pretty perplexing to the lay-person. When a mathematician writes "relatively little mathematical background," what that constitutes from their perspective is probably radically different from what a lay-person considers to be "relatively little mathematical background"

When I see "minimal mathematical background" or the like, I usually think: "Understands basic set theory, the concept of a function, and has familiarity with basic proof structures"

When I see "mathematical maturity", I think "Understands, real analysis, abstract algebra, and topology."

Hope that helps.

Re: Basic Category Theory

#45

Post by xiphias »

monktastic1 wrote: post #30

The best intro I've ever seen is Bartosz Milewski's video series (intended mostly for coders): https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI... It assumes almost no prior math knowledge (okay, you should know what sets and functions are...).

Me too...it's amazing how much I learned from it about creating composable code.

Re: Basic Category Theory

#46

Post by more_original »

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'

"There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence." -- Chen Ning Yang

Re: Basic Category Theory

#47

Post by Smaug123 »

elberto34 wrote: post #24

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.

"Stopped being about results"? I think you're referring to the building-up of the vocabulary required to attain results. As an analogy, you learnt to count (1,2,3,4,…), but then when addition and multiplication were introduced, it "stopped being about numbers, and more about abstract and complicated operations on numbers". Yes, fine, but addition and multiplication open the gateway to the study of the primes, from which most of the results of number theory follow.

Re: Basic Category Theory

#48

Post by [deleted] »

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…

[deleted]

Re: Basic Category Theory

#49

Post by wolfgke »

Bahamut wrote: post #17

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 homomo…

> 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,

This is something math and computer science students typically learn in the first two weeks in their mandatory math lectures at least at German universities

Re: Basic Category Theory

#50

Post by dualogy »

monktastic1 wrote: post #30

The best intro I've ever seen is Bartosz Milewski's video series (intended mostly for coders): https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI... It assumes almost no prior math knowledge (okay, you should know what sets and functions are...).

Looks so promising. Haven't decided yet whether I'll view through these 15 whopping hours or study the written (massive-book-length) equivalent chapters on his blog..

Sizing this up, my real question now becomes: as an active day-to-day-life programmer, by how much exactly will this propel my work if I already know and use monads, basic function composition and Haskell?

I'm sure there's some amount of skill/quality/productivity payoff if you evaluate this kind of time investment over the whole lifetime-of-work outstanding, but seems impossible to judge beforehand. Any, er, "testimonials" in terms of how this actually changed work and coding for you or others, other than "just neat geeky theory to digest and reason about endlessly"?

It's just SO MUCH material to dive into (in terms of sources lectures books etc, I'm sure once grasped everyone could condense it into a single page, though fully comprehensible only to themselves, until for teaching others once again decompressed into 15-hours-of-lectures/26-chapters-totalling-book-length) and I'm still driven by this odd old urge to "just go pump out code and make programs happen" back from my earliest Pascal/Basic days I suppose..

Post reply on HN