Another categorial week

I’m at rather a troublesome point in putting together the Gentle Introduction. That’s because I really can’t put off any more writing a couple of introductory chapters, one on ‘Structures’ and one on ‘Sets’, to say something first about the very idea of abstract structures (in some sense, the topic of a lot of modern mathematics), and then to say something about the minimal notion of set that non-set-theorists need. Neither is exactly a non-contentious topic; and in due course category theory is supposed by many to throw light on both. So it will be a balancing act, trying to do enough scene-setting but not foreclosing options. Surprise, surprise, it’s going slowly. In the meantime, here are three things I’ve learnt about category theory this week:

  1. Jean-Pierre Marquis’s From A Geometrical Point of View: A Study in the History and Philosophy of Category Theory is a disappointment. I’d tried to read this a few years ago when first getting interested in category theory, and found it tough going, so set it aside to return to when I knew more. Well, now I do know quite a bit more category theory, and more about some of the background too. And it is still tough going; too often, it is much less than perfectly clear, even impenetrable when you try to work out exactly what is being said in plain terms. I like my exposition transparent and simple-minded. So this is an opportunity lost, I think, as Marquis plainly knows a lot about a lot and has read very widely through the early days of category theory. But he’s just not getting his story across, even to this sympathetic reader coming better primed than many philosophers, I would guess. Of course, there are many potentially useful episodes, either pointing to bits of the history or suggestive of conceptual points to try to think through. However, not a successfully achieved book by my lights.
  2. Working through Leinster’s book (which is a successful book in many ways) has been continuing to be both fun and illuminating. This week, looking at the proof of the General Adjoint Functor Theorem, I learnt from Patrick Stevens how to make it all seem entirely natural: here’s his written-up version. I’ll steal from him in due course!
  3. I was quite surprised to find, in an hour’s browsing though various UK university maths departments websites, that there are almost no courses on category theory out there (indeed standard books are missing from the university library collections of the University of X and Y). Drat. Turning my notes (if I ever do) into a book isn’t going to make my fortune then! Odd though it isn’t taught more, given how combinatorially easy and how conceptually neat elementary category theory is, and how very widely basic categorial ideas are assumed elsewhere. But I suppose the first textbooks are only about fifty years old, and maths departments are notoriously conservative …

But enough procrastination. Back to thinking about what mathematicians mean by talk of structures (as opposed to what some philosophers think they ought to mean).

2 thoughts on “Another categorial week”

  1. Re 1: Have you tried Ralf Kromer’s Tool and Object: A History and Philosophy of Category Theory?

    Re 3: Is it very different with set theory? Basic set theory ideas are widely assumed elsewhere, but do many UK university maths departments have a course in set theory? It is one most maths students would take, or would they normally pick up the necessary set theory in other courses? I’m also thinking of the many textbooks that have an introductory chapter, or an appendix, that quickly covers the necessary material. Textbooks that make substantial use of category theory would presumably do something similar, or introduce the ideas as needed.

    I any case, there seems to be enough of a market that publishers offer books and shops stick them. In the local academic bookshop, the maths section currently has Leinster, Simmons, Conceptual Mathematics, and Cakes, Custard and Category Theory: Easy recipes for understanding complex maths. The closest it gets to a set theory book is Barnaby Sheppard’s The Logic of Infinity.

    1. Re 1: Kromer’s book is next on my list — watch this space!

      Re 3: Yes I guess you are indeed right about the comparative dearth of serious set theory courses too.

      Your “local academic bookshop” seems to be doing better than Heffers in Cambridge, which seems to have largely abandoned maintaining a stock in mathematics.

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