Category Theory Anyone?

I’m putting together a study group for an introduction to category theory. Who wants to join me?

Usually in the Fall and Winter, I’m concentrating on studying some semblance of abstract mathematics with a group of 20-30 kamikaze amateurs under the apt tutelage of Dr. Michael Miller through UCLA Extension. Since he doesn’t offer any classes in the Spring or Summer and we haven’t managed to talk Terence Tao into offering something interesting à la Leonard Susskind, we’re all at a loss for what to do with some of our time.

A small cohort of regulars from Miller’s class has recently taken up plowing through Howard Georgi’s Lie Algebras and Particle Physics. Though this seems very diverting to me given our work on Lie groups and algebras in the Fall and Winter, I don’t see any direct or exciting applications to anything more immediate.

Why Not Try Category Theory?

Since the death of Grothendieck I have seen a growing number of references to the area of category theory from a variety of different fronts.

Most notably, for the past year I’ve been more closely following John Baez’s Azimuth Blog which has frequent posts relating to category theory with applications I can directly use in various areas. Unfortunately I couldn’t attend his recent workshop at NIMBioS on Information and Entropy in Biological Systems, which apparently means I missed meeting Tom Leinster who recently released the textbook Basic Category Theory (Cambridge University Press, 2014). [I was already never going to forgive myself after I missed the workshop, but this fact now seems to be additional salt in the wound.]

The straw that broke the proverbial camel’s back was my serendipitously stumbling across Ilyas Khan‘s excellent post “Category Theory – the bedrock of mathematics?” while doing a Google image search for something entirely unrelated to anything remotely similar to mathematics. His discussion and the breadth of links to interesting and intriguing papers and articles within it and several colleagues thanking me for posting about it have finally forced my hand. (I also find myself wishing that he would write on a more formal basis more frequently.)

So over the past week or so, I’ve done some basic subject area searching, and I’ve picked up David I. Spivak’s book Category Theory for the Sciences (The MIT Press, 2014) to begin plowing through it.

Anyone Care to Join Me?

If you’re going to get lost and confused in the high weeds, you may as well have company, right?

Chris Aldrich

Category Theory, Anyone?

Since doing abstract math is always more fun with companions, and I know there are several out there who might be interested in some of the areas which category theory touches on, why don’t you join in? Over the coming months of Summer, let’s plot a course through the subject. I’ll suggest Spivak’s book first as it seems to be one of the most basic as well as the broadest out there in terms of applications. (There are also free copies of versions available through arXiv and MIT.) It doesn’t have a huge list of prerequisites either, so a broader category of people might be able to join in as well.

We can have occasional weekly or bi-weekly “meetings” via internet using something like Google Hangouts, Skype, or ooVoo to discuss problems and help each other out as necessary. Ideally those who join will spend at least 3 hours a week, if not more reading the text and working through problems. Following Spivak, we might try dipping into Leinster, Awody, or Mac Lane.

From the author of Category Theory for the Sciences:

This book is designed to be read by scientists and other people. It has very few mathematical prerequisites; for example, it doesn’t require calculus, linear algebra, or statistics. It starts by reintroducing the basics: What is a set? What is a function between sets?

That said, having a teacher or resident expert will be very helpful. Category theory is a “paradigm shift”—it’s a new way of looking at things. If you progress past the first few chapters, you’ll see that it’s a language for having very big thoughts and making unusually deep analogies.

To make real progress in this book (unless you’re used to reading university-level math books on your own) it will be useful to periodically check your understanding with someone who has some training in the subject. Seek out a math grad student or even a Haskell expert to help you. A growing number of people are learning basic category theory.

In order to really learn this material, a formal teacher or a professor would be best. Encourage your local university math department to offer a course in Category Theory for the Sciences. I can recommend this in good faith, because I went to special efforts to make this book available for free online. An old version of the book exists on the math arXiv, and a new MIT Press-edited version exists in HTML form on their website (see URLs below). That said, the print version, available here on Amazon and elsewhere, is much easier to read, if you want to get serious and you can afford it.

This book contains about 300 exercises and solutions. For those who wish to teach a course in the subject, there are 193 additional exercises and solutions behind a professors-only wall on the MIT Press website (see URL below). You simply have to request access.

To everyone: I hope you enjoy the book, and get a lot out of it!

Old version: arxiv.org/abs/1302.6946
HTML version: mitpress.mit.edu/books/category-theory-sciences

David Spivak, mathematician
in Description of Category Theory for the Sciences on Amazon.com

References

Awody, Steve. Category Theory (Oxford Logic Guides, #52). (Oxford University Press, 2nd Edition, 2010)

Lawvere, F. William & Schanuel, Stephen H. Conceptual Mathematics: A First Introduction to Categories. (Cambridge University Press, 2nd Edition, 2009)

Leinster, Tom. Basic Category Theory (Cambridge Studies in Advanced Mathematics, #143). (Cambridge University Press, 2014)

Mac Lane, Saunders. Categories for the Working Mathematician (Graduate Texts in Mathematics, #5). (Springer, 2nd Edition, 1998)

Spivak, David I. Category Theory for the Sciences. (The MIT Press, 2014)

Why write a new textbook on Category Theory, when we already have Mac Lane’s ‘Categories for the Working Mathematician’? Simply put, because Mac Lane’s book is for the working (and aspiring) mathematician. What is needed now, after 30 years of spreading into various other disciplines and places in the curriculum, is a book for everyone else.

Steve Awody, mathematician
on page iv of Category Theory (Oxford Logic Guides, #52). (Oxford University Press, 2nd Edition, 2010)

If you’d like to join us, please leave a comment below and be sure to include your email address in the comment form so we can touch base regarding details.