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...

easier, because it strips away the possibly confusing details and leaves the bare essentials. For instance, proving that any continuous function $f: [0,1] \to [0,1]$ must have a fixed point is a lot easier than to be given a monstrous (or even not so monstrous) function which satisfies those hypotheses and having to prove the existence of the fixed point in that particular case, because the first case is so sparse there is not much we can attempt to begin with. – Thierry Zell Nov 7 '10 at 0:23