categories

Category Theory (Oxford Logic Guides)
$155.00
This book is nice in that, once understood, it provides a quick introduction to category theory that gets to the pith of the matter right away. The author chooses illuminating examples and provides (what seems to me to be) the right amount of discussion for each topic (not too rambling, leaving the reader bored, and not to terse, leaving him confused).

However, the author was quite careless in keeping his notation and terminology comprehensible. A common problem is the reference to two different entities by the same name in situations in which it's important to distinguish between the two. For example, the author sometimes uses 'product' to mean the object that has the mapping property of the product of objects A and B and sometimes he uses it to refer to A, B, their product object, and the relevant arrows. Similarly for equalizers. Also annoying is the author's tendency to switch notation mid-proof without any notice. The author also has several unexplained uses for 'Hom(x,y)' that differ greatly.

These problems can be overcome. Once one finds out what's really meant, the text is actually quite helpful. However, I read this book on my own without access to anyone who knew anything about the theory, making it rather difficult to clarify the author's terminology.

In short, it's a book that is good on content but bad on form. I'd recommend starting with the (free) resources listed on the Wikipedia page for category theory, which, as of this writing, have less material but better presentation.

Categories for the Working Mathematician (Graduate Texts in Mathematics)
$69.95
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!

Aristotle: Categories. On Interpretation. Prior Analytics (Loeb Classical Library No. 325)
$24.00
If you're not familiar with the Loeb's, this wonderful series aims to make accessible all important Greek and Latin literature in bilingual editions - English translations with the original text on the opposite page. These books can be of great value to students of classics as well as to professionals in other fields, e.g. philosophers that are not fluent in Greek, but need an accurate and dependable translation of the works of Plato or Aristotle. And in my experience, the Loeb's rarely fail to meet expectations.

This volume contains Harold P. Cooke's translation of the Categories and De Interpretatione as well as Hugh Tredennick's translation of the Prior Analytics. I found Cooke's translations to be a little bit disapointing. The English translation often merely paraphrases Aristotle. This doesn't automatically make the translation a bad one, of course, for sometimes paraphrase is needed. But there are other translations available of these works, and, in my oppinion, Cooke's translation is inferior to J.L. Ackrill's translation of the Categories and De Interpretatione, which is both more accurate and relatively easy to read.

Now, I assume that no one would buy a Loeb primarily for the Greek or Latin text - for that you would turn to the Oxford Classical Texts or other critical text editions. So if you're buying a Loeb it's either for the translation or to be able to compare an English translation with the original. If you need to compare an English translation of these particular works with the Greek text, then this volume will be useful to you. However, if you just want to read these works in translation, you might very well be satisfied with this one, but I still recommend other translations such as J.L. Ackrill's excellent translation of the Categories and De Interpretatione.

Categories On the Beauty of Physics : Essential Physics Concepts and Their Companions in Art and Literature
$24.95
This is a great concept: illustrate principles of physics using art and nature. Unfortunately, the execution is poor. The book's layout is dull, and the whole thing comes across as boring rather than fascinating. I had hoped to share it with someone who doesn't like science, but couldn't, because it would have turned him off even more. A great disappointment.