A Concrete Approach to Category Theory

There was a time,not so long ago,when you really couldn't-at least not in any great depth.This was because most of the important sources were quite advanced graduate level monographs that presumed at least a first year graduate student's knowledge of topology and algebra. MacLane's treatise, of course,is of this nature. So is Herrlich/Strecker (which I actually like better in some ways).

The main counterexamples to this rule were the advanced undergraduate/graduate level textbooks in algebra and topology that taught basic category theory concurrently with the material they were trying to teach. Good examples of this were MacLane/Birkoff's Algebra and Ronald Brown's excellent Topology And Groupoids. But these sources really don't cover category theory in great depth-they just cover what's needed to understand a categorical/homological diagram approach to their subjects.

It seems to me what you're really asking is whether or not one should try and learn abstract category theory independently of its motivating examples. This question is a good one and it's been a matter of great debate here and on the companion board, Math Overflow. In fact, I asked the question there last year and got some terrific feedback from a number of people.You can read my comments there-my opinion really hasn't changed on it:

https://mathoverflow.net/questions/41057/categories-first-or-categories-last-in-basic-algebra

To quote myself from that board: I've never really been comfortable with category theory. It's always seemed to me that giving up elements and dealing with objects that are knowable only up to isomorphism was a huge leap of faith that modern mathematics should be beyond. But I've tried to be a good mathematican and learn it for my own good. The fact I'm deeply interested in algebra makes this more of a priority..........

A number of my fellow graduate students think set theory should be abandoned altogether and thrown in the same bin with Newtonian infinitesimals (nonstandard constructions not withstanding) and think all students should learn category theory before learning anything else. Personally, I think category theory would be utterly mysterious to students without a considerable stock of examples to draw from. Categories and universal properties are vast generalizations of huge numbers of not only concrete examples,but certain theorums as well. As such, I believe it's much better learned after gaining a considerable fascility with mathematics-after at the very least, undergraduate courses in topology and algebra.

To this, I'll add a lot of people tell me my attitude is antiquated and that the majority of mathematics can and should be rephrased in terms of categorical constructs from the beginning. My reply is basically the above with the following added caveat: You also drive with your feet, that doesn't make it a good idea..........

This post has gone on too long,but in closing, I will say that there is now an excellent source for introducing category theory to undergraduates while not watering down the subject and simultaneously providing many good examples: Category Theory by Steven Awodey. This is the only book I would consider using to teach the subject to undergraduates. It's quite pricey, but it's now in paperback, which is a bit less expensive. Definitely worth the cover price if you're serious about category theory.

Yes, you just learn the category theory. Presumably what you mean, is that when reading Mac Lane or Herrlich/Strecker it seems as though you need to understand what $\mathbf{Grp,Rng,Ring,Top,Toph},R\text{-}\mathbf{Mod},\mathbf{Set},\mathbf{Ban},\mathbf{BooAlg}$,... means. Well, you don't really, but is seriously helps. In other fields one needs concrete examples to tests one's intuition, in category theory, one needs other fields. So, sure you can learn category theory, but it will undoubtedly be rather dry and meaningless if you don't know some of the basic examples of (concrete) categories. If you are trying to get right to the category theory though, there are definitely "more important" examples than others. You should know $R\text{-}\mathbf{Mod}$ because category theory is so prevalent there, it's probably the richest source of applications. The same could be said of $\mathbf{Top}$, especially for the very well-known functors out of it. You should know $\mathbf{Field}$ because it is a good source of counterexamples (no terminal object, just to start, but it also lacks a lot of nice constructions). You definitely need to know $\mathbf{Set}$, but I hardly doubt that's a a problem (keep $\mathbf{Set}_\ast$ pointed sets in your back pocket, another good counterexample, and easy to understand).

I must strongly disagree with most of the answers here. Lawvere/Schanuel introduces CT via (di)graphs instead of "algebraic" objects. This means it's basically on the Plato's-slave level of naive, visceral understanding, like set theory or high-school geometry. It also means it would be a wise read for even experienced mathematicians to really pump their intuitions about the subject. And, while it's indeed been used on actual high-school kids (the curious types who are often given set theory in your fancier schools), it will take you right up to adjunctions!

After that, Lawvere has written a more advanced book with Rosebrugh that is based on the category of sets, another worthwhile intuition pump even for the more advanced. But other than that, it's true that after Lawvere/Schanuel there's no real literature for you beyond that if you don't know algebra. Fortunately, you can take care of that quickly. Don't get scared by hard books like Lang or dry, matrix-heavy ones like Artin; pick up a Dover copy of Pinter (it's easy and fun!), learn a little linear algebra (yeah, you pretty much have to), be willing to keep learning a little more algebra from easy sources (e.g. internet stuff) even as you proceed with that CT you're so impatient to get to, and above all keep in mind the general idea of a structure, over and above a set, and what it would be for a morphism to preserve it. But yeah, you're ready for your first crack at Mac Lane! He's actually a really gentle writer, and the good thing about this subject is you can easily just use the examples you do understand for now, though of course the intuition comes easier the more you can relate to.

In sum: There is a way: Lawvere/Schanuel, one of the best books anyone will ever read. And I think it would be a tragedy if someone who's so excited and inspired by CT's abstract beauty were to be discouraged by all these dour warnings in an all-too-familiar "pay your dues kid; this stuff's a long and hard journey" spirit. Especially since algebra is so often made to seem so boring. (Really, it's not! And let CT be your motivation to uncover that particular pleasant surprise.)