I don’t know what “inverse problem theory” is, so I can’t help you with that example. In physics the power of category theory is by now well-established: a bunch of physical systems are described by ‘modular tensor categories’, and this way of thinking about them is crucial for understanding things like the quantum Hall effect, anyons, etc. I also believe that quantum theory and gravity can only be unified by seeing their relationship with the help of categories. Category theory is also widely recognized to be important in computer science, e.g. the use of monads in Haskell, the relation of the lambda-calculus to cartesian closed categories, and the like.

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.

]]>Here’s a better link:

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

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

]]>Thanks for the link! You can’t use Haskell without using monads. More generally, there’s a kind of computer scientist who uses lots of ideas from category theory, and that’s one reason I like talking to computer scientists.

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….

]]>Here’s an interesting exposition ( http://bartoszmilewski.com/category/category-theory/ ) by Bartosz Milewski that introduces category theory and goes on to derive Yoneda’s lemma using numerous examples to illustrate important concepts employed. Most of the examples involve simple Haskell programs and the diagrams include piggies, sacks of potatoes and fireworks (I kid you not!), presumably inspired by the lessons the author used to teach his 10-year old son about categories. As to practical applications of category theory, it is claimed that anyone who wants to understand implementations of Haskell or be involved in its extensions must have a basic understanding of category theory. Amazing!

]]>The category theory was helpful, but largely because (already being familiar with category theory) I already wanted to know more about the category of differentiable manifolds qua a category. (In particular, I really needed to know about it qua a site for my doctoral research, and reading Lang helped orient my thoughts properly even where it didn’t directly give me the answers that I needed.) To say more, I would really have to look at the book again. (I might have a copy downstairs.)

]]>Thanks, Toby. I was able to do a quick scan on Amazon of S. Lang’s Introduction to Differentiable Manifolds (2002), which seems to be an update of his earlier text and apparently limited to finite-dimensional manifolds. There is a 4-page introduction to category theory at the front of the book, and frequent references to category concepts in the body of the book. But I agree with your assessment – the book is definitely too advanced for beginners. However it does seem to offer an opportunity to assess the value of what category theory brings to the description of a mathematical topic. My question to you, Toby, or to someone with your background, would be, to what extent (whatever that means) does the use of category theory in this book aid in the presentation and explanation of the concepts of differentiable manifolds? Or put differently, if all the category theory terminology were removed, would the book become significantly less useful? I realize this is pretty vague, but at least it is a fairly narrowly-defined problem that might give some specific insight into the value of using category theory as a tool to describe and teach mathematical concepts.

]]>Charlie Clingen wrote eventually:

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.

]]>Charlie wrote:

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.