"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 identities. ;-)
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.
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.
"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
Bob Coecke writes good introductions to category theory, you might find some of his stuff enjoyable. Although he is often also introducing quantum physics and linguistics at the same time.
The "note to the reader" clarifies things a bit -- he's aiming for requiring "no more mathematical knowledge than might be acquired from an undergraduate degree at an ordinary British university". Though he does not specify whether he has in mind a mathematics degree, I think this can be deduced from the fact that he indicates that the text developed out of a master's-level course.
The text developed from approximately six lectures' worth of the MMath-level 24-lecture Part III Introduction to Category Theory at Cambridge, I believe. (Source: I took that course last year, and was part of a small reading group studying Leinster's Basic Category Theory at that time. We found that book really, really helpful.)
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 computer programming, or computer science in general? I find this really interesting from an outsiders view and want to learn it. I'm looking for a truly basic and gentle approach to learning category theory, without removing the rigor.
The target audience of this book isn't the layman, it's people studying mathematics. So a phrase like "relatively little mathematical background" is meant in the context of academia.
I don't think there's a basic and gentle approach to learning category theory that doesn't remove the rigor. You can learn at a high level what some of the stuff in category theory and get a basic feel and intuition for it, if that's what you want. But, if you want to learn category with the rigor, you're going to have to first learn how to write a proof in mathematics. Really though, you're going to want to have a decent grasp on some basic abstract algebra. Otherwise you're never going to be able to understand the examples for category theory.
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".
"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.
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.
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.
I think it's easily done when someone knows a topic too well to teach it to beginners from outside the field. Rings, fields and identities may just be math's equivalents of string, operator, or variable, when similarly used by introductory programming books without any explanation.
Well, everything has some amount of implied/customary prerequisites. You wouldn't complain if a books "Introduction to Metaprogramming in Ruby" or "Introduction to C++ templates" didn't teach you programming.
In math many "general topics" like calculus, linear algebra, probability etc are sometimes studied a few times (eg the second course in calculus might start from scratch and carefully construct the real numbers, which the first course skipped), and I've noticed some authors use "Introduction to X" to mean "this textbook is meant to be a first course in X"
It feels like "relatively little" to many authors means "a complete set of undergrad math courses from a fairly math-heavy major" (which could be something like calculus including vectors, differential equations, linear algebra, real analysis, and maybe abstract algebra and discrete math). Maybe we need another term for "OK with logic, proofs, and calculations and doesn't mind learning notation, but hasn't had a lot of university-level courses, or in any event doesn't remember them".
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.
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.
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.
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."
jargon like "homoporhism, rings, multiplicative identity" are just scary sounding words for simple concepts. Especially this day and age with wikipedia.
Now, not to say that proving things about groups and rings is simple. but just because the word has 5 syllables doesn't mean the concept is hard. Whereas certain concepts in calculus - I think - are quite difficult to solve for even simple looking integrals.
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 identities. ;-)