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'
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.
This paper is fascinating:
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.
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.
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 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.
If you read the referenced book, you'd know the author does exactly this.
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.
The book doesn't really require much knowledge.
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"
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.
I think you really need no more than high school algebra to understand basically what category theory is about.
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.
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. ;-)