So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I started to learn about the categorical way of thinking.
I think that the deal is as follows: One begins to gain all sorts of intuition about, for instance, groups when one realizes that it doesn't pay to think about elements of a group nearly as much as it does to think about morphisms to/from a group. The category-theoretic point of view is a tool that lets you gain intuition by moving up and down the hierarchy of abstraction. Maybe I'm not articulating this that clearly, but hopefully you've had similar experiences.
The problem is, I don't know any way to get a handle on the categorical way of thinking without learning category theory, and I don't know any way of learning category theory without wading through tons of abstract nonsense before you can begin to understand why it's valuable. (This is an even worse problem if, like my friend, you don't have any interest in the motivating examples from topology or geometry.)
So, what can I recommend that might help my friend start thinking categorically without drowning him in a sea of abstraction? Or is the "algebra sucks" phase a necessary stage of mathematical development?
ETA: Just to be clear, this is mostly undergrad-level stuff we're dealing with, so while I'm not opposed to easy ways to motivate category theory or get someone hooked, the fewer prerequisites the better…
Best Answer
There are two books that I would highly recommend to any one (especially a high school student or an undergrad with some mathematical maturity) wanting to learn category theory (CT) without necessarily learning all the "abstract nonsense." That is, it is possible to become thoroughly acquainted with a categorical way of thinking, if you will, without learning all the definitions beforehand. The books are "Conceptual Mathematics: A First Introduction to Categories" by Lawvere and Schanuel, the second being "Sets for Mathematics" by Lawvere and Rosebrugh.
The first book (Conceptual Mathematics) is, I believe, the best introduction to CT via sets and functions. From a pedagogical point of view, it is very well-written with lots of examples that slowly but surely develop the diligent student's category-theoretic intuitions. The notion of a functor is only mentioned in the middle of the book and that too in the context of a monoid (which is a very nice concept to learn) and that the notion of a natural transformation isn't even mentioned in the book. Indeed, universal mapping properties are introduced two-thirds of the way into the book. However, that does not prevent Lawvere (and Schanuel) from talking about a lot of stuff. One is surprised by how much CT one learns by the time one finishes reading (and solving the exercises in) the book.
The second book (Sets for Mathematics) essentially is a superb exposition on the elementary theory of the category of sets, and the book can be read either after finishing the first book (mentioned above) or in conjunction with it after one has gained some proficiency in understanding and writing (simple) proofs in CT.
Anyway, if your friend really wants to learn CT without getting "bogged down" by all the definitions, then Conceptual Mathematics is the book to read and absorb. The book looks deceptively easy and devoid of a lot of the standard category theoretic content, but nothing could be further from the truth!
Hope this helps.