MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Does category theory help understanding abstract algebra?

I'm studying category theory now as a "scientific initiation" (a program in Brazil where you study some subjects not commonly seen by a undergrad), but as I've never studied abstract algebra before, so it's hard to understand most examples and to actually do most of the exercises. (I'm using Mac Lane's Categories for the Working Mathematician and Pareigis Categories and functors.)

To solve this, my advisor recommended me to get S. Lang's Algebra as a reference, but I don't know if that's the most appropriate book and if it's better to get Lang and study algebra through category theory or to study (with a different book and approach, maybe Fraleigh) algebra and then category theory.

PS: I'll have to study by myself (with my advisor's help), as I can't enroll in the abstract algebra course without arithmetic number theory.

-
Studying category theory without a fair amount of prior experience with abstract algebra is like studying calculus without knowing how to graph a linear equation: it makes no sense. There are plenty of concrete topics in elementary abstract algebra that are not in the "standard curriculum" and don't need too much preparation and are a lot more "real" than category theory. For example, classification of finite groups of rigid motions of 3-dimensional Euclidean space; needs just basic group theory and linear algebra. Someone who knows your background should suggest an alternative topic. – BCnrd May 2 2010 at 0:11
huhm? abstract algebra is by far not the only source for motivating category theory, including proofs, constructions and notations. – Martin Brandenburg May 2 2010 at 0:26
The book I used in my first abstract algebra course was Dummit and Foote, which I thought was very well-written. I think I would go so far as to say that it was the best introductory math book that I've used. – Peter Samuelson May 2 2010 at 1:22
The "Location" field on your profile and your use of the term "scientific initiation" makes me pretty sure that you attend the same mathematics department that I did. And your advisor's recommendation -- that somebody's first contact with algebra should be a graduate textbook by Lang -- makes me all but sure who your advisor is. My advice to you is to take your advisor's advice with hefty doses of salt. Presuming I'm correct in my guess, he has very good taste and knows a lot of math, but he doesn't always set realistic goals for his students. – Pietro KC May 2 2010 at 1:25
Sure, category theory helps in understanding abstract algebra, especially graduate algebra. But you know what is even more helpful in understanding graduate algebra? Undergraduate algebra. Learning some categorical language in your first (undergrad) algebra course is not as crazy as some might think (this was in fact the perspective taken in the undergrad algebra course I took at U. Chicago; it worked out okay), but learning category theory before you learn undergraduate algebra is about as far-fetched as any study plan I have ever heard of. – Pete L. Clark May 2 2010 at 2:02

Sounds like you might want to petition for an exception to the prerequisites. I don't think Lang's Abstract Algebra is probably your best bet (stick to something decidedly undergraduate -- maybe Gallian?), nor do I think that trying to digest either all of abstract algebra or all of category theory is your best bet. I'd aim for one major result in abstract algebra which has an analogous statement in a variety of other categories, and then see what carries over to the category-theoretic framework. One idea would be to understand the classification of finite abelian groups in your abstract algebra work, and try to understand how the result and the proof techniques carry over/generalize.

p.s. The answer to the title question is definitely yes. :)

Edit: Let me add on what I think is almost certainly the best place for you to start on categories, which is Lawvere and Schanuel's "Conceptual Mathematics: A First Introduction to Categories" (double edit: which I see mathphysicist also listed). In fact, with this book in mind, it's actually the abstract algebra part of your project that now sounds the most daunting -- is that negotiable? Their discussion of Brouwer's fixed-point theorem would make an excellent topic.

-
 I'm very surprised there are still mathematics programs across the world where undergraduates don't learn abstract algebra,Carn. Even a small amount-even a group theory course for undergraduates.That's a HUGE obstacle for any undergraduate preparing for graduate studies to overcome even with an instructor's help. – Andrew L May 2 2010 at 2:04 Andrew, I think you misunderstood the question. The point is not that algebra is not part of the undergrad curriculum, but rather that the OP is just starting his undergrad and hasn't gotten to it yet. What his sci-init covers, and is not part of the usual curriculum, is a substantial amount of category theory. His advisor, if I guessed right above, works in an area which makes extensive use of it. The OP's curriculum includes a semester course in group theory and another which covers some commutative algebra plus fields and Galois theory. – Pietro KC May 2 2010 at 4:22 @Pietro Ok,gotcha. But still asking a lot of your student. LANG as a reference for just learning algebra?!? This advisor either went to Yale when he was 17 or just doesn't care much..... – Andrew L May 2 2010 at 7:02

I can recommend several much better sources that will ease your transition into both abstract algebra and category theory, Silva.

Lang is far too difficult for a first brush with abstract algebra-and MacLane is even MORE difficult for a neophyte in algebra. Category theory has VERY far reaching conceptual implications for most of modern mathematics,not just algebra.So no,in principle,you don't have to learn abstract algebra to learn it-but that's where most mathematicians have infused it.This is because it's natural to organize types of structures into categories and that's really what algebra is all about:types of structures i.e. sets with binary relations on them.

There are a legion of great abstract algebra texts,but my favorite is E.B.Vinberg's A COURSE IN ALGEBRA,available through the AMS. It takes a very concrete,geometric approach and builds an extraordinary amount of algebra from first principles all the way up to the elements of commutative algebra, Lie algebras and groups and multilinear algebra.It will help you learn a great deal of algebra very quickly and without the confusion of learning category theory simultaneously. Another geometrically flavored-but a bit more challenging-book is the classic ALGEBRA by Micheal Artin. Indeed,I think the 2 books compliment each other very nicely. Mastering both books will give you a very good working knowledge of algebra and you'll be more then ready to tackle Lang's book after that.

As for category theory,the best introductory text I know is CATEGORY THEORY by Steven Awodey. Gentle,rigorous and masterly,it's the best book for undergraduates and the only one I'd use for a beginning course in category theory for students that don't have strong backgrounds in algebra. It's pricey,but totally worth it. One other very good-and short-book you should look for and I heartily recommend is T.S.Blyth's CATEGORIES-a terrific short introduction for any student with good mathematics background that wants just the basics in category theory. It's REALLY hard to find now,but if you can get a copy,by all means,do so.

-
Vinberg's indeed pretty good and readable. As for categories per se, see two other books in my answer in addition to Awodey. – mathphysicist May 2 2010 at 0:29

You can also give a try to other books on category theory which are more accessible than, say, the MacLane's classics.

Here they go:

1. Conceptual mathematics: a first introduction to categories by Lawvere and Schanuel

2. Arrows, structures, and functors: the categorical imperative by Arbib and Manes

3. Category Theory by Awodey

Lawvere & Schanuel require almost no math background at all, Arbib & Manes and Awodey are somewhat more advanced but should be at least partially available to a math undergraduate without much knowledge of abstract algebra.

-
Lawvere and Rosebrugh's Sets for Mathematics, is also a great read. It has set theory axioms interspersed throughout, but works nicely as an introduction to categories. It's a very gentle read. – AndrewLMarshall May 2 2010 at 4:56
Lawvere and Schanuel is amazing; I second the recommendation. – Qiaochu Yuan May 2 2010 at 6:33
Awodey is excellent and should be well within reach. It is often tackled by computer scientists and logicians with minimal (or even no) knowledge of algebra. – Justin Hilburn May 2 2010 at 21:36
A softcover version of Awodey's book is supposedly becoming available this month. It's less than half the price of the hardcover version. – supercooldave May 15 2010 at 13:37
@supercooldave FINALLY,AMEN. – Andrew L Jun 9 2010 at 4:51

Silva, you're studying category theory way too early. You don't have a background yet that can give you an appreciation for the point of what you're being asked to understand, so probably at best you can follow things line by line (maybe not even that much?) but can't get anything like a bird's eye view of the point of it all. This is like trying to teach abstract linear algebra to someone who hasn't yet had any high school algebra. The motivation is nowhere to be found.

Ask your advisor what he considers to be some of the important inspiring examples for category theory. If you don't understand what those examples are, that's a pretty concrete illustration that something is wrong (but then it seems like you already realize it). Then go speak to someone else who can suggest other topics more closely aligned with your background or that start at a more basic level.

To answer the question, yes category theory gives a lot of insight into the nature of abstract algebra, but only after you've studied enough of the subject on its own for certain basic intuitions (like the meaning and significance of kernel or quotient constructions) to be in your head first.

-

There is the book Algebra: Chapter 0 by Paolo Aluffi that might fit your needs. It is a textbook on algebra (as the title says), but it uses the language of category theory from the beginning. Category theory is mostly used to motivate definitions using universality properties.

It is only in the last two chapters that the author introduces more advanced concepts from category theory (functors, abelian categories, etc.).

-
Already mentioned it,Alexander. It looks pretty good,but I personally think it's a gentle graduate level text. It would be pretty hard for someone without much mathematical training to use. – Andrew L May 2 2010 at 6:59
If you mentioned it already, I'm sorry. I cannot find a reference to the book in your answer, however. The text is designed for graduate students, but I think that it is readable for undergraduates as well. I used this book to learn abstract algebra for the first time, and at least for me it worked. – Alexander Noll May 2 2010 at 11:52
+1 I like Aluffi's book very much. I would have been glad to have known it earlier. I like Goldblatt's "Topoi", too. (Maybe not quite adequate because topoi are related more to geometry, set theory and logic than to algebra. Nevertheless, I found it a gentle (general) introduction.) – Hans Stricker Sep 13 2011 at 15:02