Have you ever tried reading Descartes' "Geometry"? It's not a good place to learn about coordinate geometry. I tried. This was almost 10 years ago, but I still remember it pretty well. Ok, so maybe the experience was even a bit traumatic.



Usually when someone works out a theory, it takes a fresh perspective (or two, or ... you get it) to really digest it, and come up with a reasonable way of teaching it to newcomers. It's less evident nowadays, with improved communications technology and such, but people aren't exactly turning to Grothendieck's expositions as their intro to his geometry either. Mac Lane is an exception.



This book seems completely inapproachable. The title is scary. The topic is scary. Open to a random page and try to judge its accessibility: scary. Well, here's the real story: you need to know algebra through modules, and it'd be nice if this algebra background introduced "universals" like abelianization or free modules in a way that involved the diagrams and the unique mappings you get from the given ones. If this stuff makes any sense, you can read this book. It's not that scary. If you're up to the challenge, you might even enjoy it. This is actually my favorite book.



Here's the approach that I feel worked well for me:



- gloss over the set-theoretic foundations at first. Make sure you know the proper class/set and large/small category distinctions, but don't dwell on them much.



- focus on the examples that are familiar, but read through the others too. Mac Lane uses tons of examples to suit a variety of backgrounds, and his presentation is so clear that the theory can often explain the examples.



- trust the author. It may seem like product or comma categories deserve fuller treatment with more motivation. No. Let Mac Lane's 'minimalism' infect your thinking: it's no more complicated than what's on those pages. Make sure you *know* what's there, and you will come to *understand* the material as it is fleshed out through exercises or later writing.



The last point has been the most important for me. This book has been a great lesson in clear thinking, which is of extreme importance in mathematics. Why? It's complicated enough!