In other branches of science categories have not yet proved their worth, but I’m personally convinced that this is mainly because nobody has put a sustained effort into applying them there. That’s what I’m working on now.

]]>http://bartoszmilewski.com/2013/05/15/understanding-yoneda/

It goes directly to the Understanding Yoneda tutorial. He has a posting on monads too.

]]>I hope Bartosz Milewski explains the Yoneda Lemma in ways that make it accessible to more people. It’s very simple in a way, and very important, but it seems to require a lot of work for most people to understand it and (harder) appreciate its applications. That was certainly true for me. I have a couple of students now who need this lemma in their work on network theory, so I’ve been seeing how quickly and easily I can get them to understand it. I have certain pedagogical tricks that seem to work….

]]>Lang, S. (1972), Differential manifolds, Addison Wesley, London. I haven’t had the opportunity to check Lang. Is anyone here familiar with that book, and do you have an opinion as to how effective the use of category theory is as an expository technique in this book?

I’ve read that book and like it. I already knew category theory and differential topology, so it’s hard to say how well it did at explaining those to beginners. As I recall, the category theory was gentle, but the differential topology was advanced, with infinite-dimensional manifolds almost from the start.

]]>In “Modern Differential Geometry for Physicists”, Second Edition, p79, Chris J. Isham comments that category theory is the natural language to describe many of the constructions made in differential geometry, but then goes on to say he will not introduce such ideas formally in the book.

Interesting! That’s a great book, but I hadn’t remembered that remark.

When I first met Isham, he was famous for being a very well-informed critic of all known approaches to quantum gravity. He thought none of these approaches were sufficiently radical. Later, he went on to develop an approach to quantum physics based on category theory—or more specifically, the branch of category theory called topos theory, which takes the category of sets and functions and generalizes it to ‘topoi’, which are categories in which you can do *all of mathematics*, just as you can do all mathematics with sets and functions. Here’s a nontechnical intro to this line of thought:

• Chris Isham, Topos methods in the foundations of physics.

I haven’t had the opportunity to check Lang. Is anyone here familiar with that book, and do you have an opinion as to how effective the use of category theory is as an expository technique in this book?

I’m not familiar with that book, though I’m familiar with Lang: when I was a postdoc, he was often the only other guy in the math department at 2 am. He wrote a book a year, apparently while he took a summer cruise across the Atlantic. I learned complex analysis as an undergrad from his book *Complex Analysis*, and I liked that book a lot.