Category theory

Telling your monos from your epis

Reposting from many many moons ago ….

Ok, so how do you remember which are the epimorphisms, which are the monomorphisms, and which way around the funny arrows get used?

Since the textbooks don’t seem eager to offer helpful mnemonics, I offer a forgetful world the following: go by alphabetical proximity!

L-for-left goes with M-for-mono, and P-almost-for-epi goes almost next to R-for-right. OK?

But what does that mean? Simple.

A mono is of course a left-cancellable morphism, and you signal one using an arrow with an extra decoration (a tail) on the left.

Dually, an epi is a right-cancellable morphism, and you signal one of those using an arrow with an extra decoration (another head) on the right.

Easy, huh? Well, it works for me — and these days, I’m grateful for all the props I can get. You can thank me later.

Category Theory, without too many tears?

So, I’ve pressed the “publish this” button on the Amazon KDP system for Beginning Mathematical Logic, and will let the world know when it gets through the review process and goes live.

And for my next Big Red Logic Book? A few years back I put together some notes on category theory (running to almost 300 pages). And despite their very rackety half-baked form, they are downloaded startlingly often — almost a thousand times in January.  Ye gods! So it’s decision time. Do I let them continue to stand as they are, despite their unsatisfactory, unfinished, form? Or do I try to make them more respectable, and round them out into a more polished book form?

On the one hand, I’m really pretty embarrassed to leave the notes online in their current state. On the other hand, having putting the work in earlier, I’m reluctant to trash them. Which leaves the remaining option, of getting down to more work and and making a better fist of it. So here goes …

I think I’ll take it in three phases. First, go through the current version, correcting all the typos and thinkos I’ve been told about, improving the presentation wherever I can, to make a better version of the existing  material. Second, get back to doing quite a lot more reading and rereading in category theory, so I can round out the existing material e.g. with more examples, more applications, and additional closely related ideas at the same level. Then third, decide which further topics (if any) I should add at the end to make a more satisfying book (though given the current length, I wouldn’t want the book to extend its reach very far into new territory).

That should all take quite a while, but (I hope) not too many tears. Watch this space for occasional updates.

Book note: Ralf Krömer’s Tool and Object

It is inevitably going to be difficult to write illuminatingly about the history of category theory. For this is entangled with the distinctly complicated history of mid-twentieth-century topology. Colin McLarty sees the difficulty like this:

For even a rough understanding of [just some of] the problems [topologists] faced we would have to go into the array of homology theories at the time and the forefront of 1940s abstract algebra, and we would do this without using category theory, and we would waste a lot of time on things category theory has now made much easier. We could give a few trivial examples just before reversing the order of discovery to define categories, functors, and natural transformations but precisely the examples serious enough to have motivated the definitions are too hard to be worth giving now without benefit of categorical hindsight.

Is that too pessimistic? Well, this much is surely true. It would take rather exceptional expositional skills, combined with an exceptional depth of mathematical understanding, to be able to helpfully isolate and explore critical moments in the development of category theory, while doing this in a way that is both interestingly detailed and yet also still quite widely accessible.

For different reasons, it is a challenge too to write illuminatingly about the philosophy of category theory. Working out what is really novel about categorical concepts and approaches isn’t easy. Working out in what senses category theory does or does not provide a new kind of foundation for mathematics isn’t easy. And we are not exactly  helped by the fact that some category theorists are wont to make distractingly sweeping claims about the philosophical significance of what they are up to, claims which are hard to deconstruct. So it would take a different set of skills, beginning with a serious feel for the philosophy of mathematics more generally, to tackle the philosophy of category theory.

It is highly ambitious of anyone, then, to take on writing a book which is intended to be both ‘A History and Philosophy of Category Theory’. But that’s the subtitle of Ralf Krömer’s 2007 book Tool and Object. This has been on my ‘must read one day’ list for quite a while. I’ve at last had time to take a serious look at it. How well does Krömer succeed at the daunting dual task?

I found the book a very considerable disappointment, even allowing for the difficulties we’ve just mentioned. Life being short and all that, I’ve decided against a chapter-by-chapter commentary here, as it would take a lot more time than it would be worth, either for me as writer or for you as reader. But in headline terms, the philosophical bits are just far too arm-waving for someone of my analytic tastes; and I found the historical mathematical exposition just too unhelpful, even for someone coming to the party with a decent amount of mathematical background. The exception, perhaps, is Chapter 6, ‘Categories as sets: problems and solutions’ which is more closely focused on one familar issue, and is quite a useful guide to some of the discussions on “The possibilities and problems attendant on the construction of a set-theoretical foundation for CT and the relevance of such foundations”. And forgive me if I leave it at that. Your mileage may vary of course; but I can’t recommend the rest of this book.

Category theory and quantum mechanics

My last link to something categorical turned out to be pointing to a less-than-splended online resource. I hope this is rather better!

I’d heard tell of people interested in quantum foundations and quantum information getting entangled (see what I did there?) with category theory. And by chance, I  stumbled a few days ago across details of a course currently being run in Oxford. The course materials are a late draft of Categorical Quantum Mechanics by Chris Heunen and Jamie Vicary which has recently in fact been published as a book by OUP. This strikes me (in contrast I fear to that book I mentioned by Fong and Spivak) as extremely lucid and well-organized; and you don’t in fact have to read very far to see why quantum theorists might indeed be interested in monoidal categories as a mathematical tool. My QM is very very rusty; but if you have a smidgin of knowledge, this does seem worth dipping into, if only to get a glimpse from the sidelines about what the cool kids are up to …

Programming with categories, again

Well, I started watching the on-going series of lectures which I’d linked to a couple of blogposts ago. Frankly I can’t say the early lectures are at all good, and can’t now recommend you try them. They all seem, to put it kindly, very underprepared, underpowered, undercooked, low in nutritious content! Maybe later lectures are better.

I’d say more, but I’m saved the effort because there’s now a long comment to that earlier post sent by Peter F. which well expresses some of my concerns about the whole thing.

Programming with categories

One of the books I’d set aside until the main work on IFL2 was done and dusted is Brendan Fong and David Spivak’s An Invitation to Applied Category Theory (CUP, 2019 — there’s a late version freely available here). I’ve now had a first proper look. I can’t say that the book overall works very well for me. When the authors are talking about things I already know a fair bit about, I find there are indeed interesting approaches and illuminating comparisons. But when they are introducing new-to-me material it, all goes by too fast to be very helpful. That could of course just be a comment on my inadequacies as a reader! — but I’d be quite surprised if mine weren’t a common reaction. Chapter 7 on sheaves and toposes, for example, is surely going to pretty impenetrable to someone who doesn’t already have some handle on this stuff.

However, there are obviously lively minds at work here, and so I’m encouraged to take a look at another of their projects — an MIT course of lectures with the estimable Bartosz Milewski (whose Category Theory for Programmers I think is terrific). The course, Programming for Categories, has a web page here with planned links to videos of each lecture and also links to lecture notes. The lectures are just starting, and new videos/notes should appear five times a week for four weeks. I’ll hope to learn a lot!

Back to categories

IFL2 is off to the proof-reader. So, for some weeks, time to think of other things.

A while back, I constructed a webpage linking to online materials on category theory at an introductory/middling  level, including lecture notes, (legally available!) books, and videos of lectures, more than 50 items altogether. Here it is!

I initially wrote this really for my own use, to keep track of things I found. But the page has got over 25K visits in the last year, so obviously some others are finding it useful too.

This page of links hasn’t had much attention in recent months while IFL2 was occupying my mind. However, I’ve just now updated it, removed a couple of dead links and corrected some others. Please do let me know of appropriate recent materials I should be adding to the list — and do spread the word to students who might find it useful.

Category theory page updated

I’m not really keeping abreast with what’s available on elementary category theory right now — who would have thought that revising an elementary logic text would be so all-consuming? (Maybe it is one of those cheering features of, erm, mature years … you can only think about one thing at a time.)

However, since I was just about to bring the resource to the attention of our local Part III Maths students, I thought I should quickly check my category theory page here – which, if you don’t know it, links to a lot of available lecture notes, legally available books etc, not to mention my stalled Gentle Intro to Category Theory. So I’ve updated some links, deleted some other seemingly dead ones, and done a small amount of tidying. Do let me know about any errors and omissions and about any newly available lecture notes, etc.

Lots of fun reading if you like that kind of thing …

Category Theory: A Gentle Introduction

At long last, I have updated my  notes Category Theory: A Gentle Introduction (now some x + 291 pages).

A good while ago, I received lists of corrections from a number of people, and just recently I’ve had another tranche of corrections, making over a hundred in all. Mostly these corrections noted typos. But there were also enough mislabelled diagrams, fumbled notation mid-proof,  etc., to have no doubt caused some head-scratching. So I can only apologize for the delay in making the corrections.

I have also added a new early chapter and restored a couple of sections that were in a rather earlier version but got lost in the last one (thereby breaking some cross-references and no doubt producing more head-scratching).

These notes were originally written for my own satisfaction, trying to get some basics clear. But I know some people have found them useful (despite their very obvious shortcomings, unevenness,  and half-finished character). So I hope some others will find the update helpful: you can download it from the category theory page here.

The Language of Category Theory

I’m taking a week or so off from on working the d****d second edition of my logic text (it’s quite fun, if you like that sort of thing, most of the time: but it is good to take a break). I’m instead updating, just a little, my Gentle Intro to Category Theory, about which more when the revised version is ready for prime time (within the week, I hope). So I’ve now had an opportunity to take a quick look at Steven Roman’s An Introduction to the Language of Category Theory (Birkhäuser, 2017) which in fact has been out a whole year.

This book is advertised as one thing, but is actually something rather different. According to the blurb “This textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible.” We might, then, expect something rather discursive, with a good amount of the kind of informal motivational classroom chat that is woven into a good lecture course and which can be missing from a conventionally structured textbook. But what we get is actually much closer to a brisk set of lecture notes. For the book travels a long way — through the usual introductory menu of categories, functors, natural transformations, universality, adjunctions (as far as Freyd’s Adjoint Functor Theorem) — and all in just 143 pages before we get to answers to exercises. Moreover, these pages are set rather spaciously, with relatively few lines to the typical page. So certainly there isn’t much room for discursive commentary.

And I would have thought that the sequencing of topics would leave floundering some of those who would appreciate a gentler introduction. So we get to the Yoneda Lemma long before we eventually meet e.g. products (and that as part of a general treatment of limit cones). Yet aren’t products a very nice topic to meet quite early on?  — in talking about them, we  explain why it is rather natural not to care about what product-objects are intrinsically (so to speak) but rather natural  to care instead about how the product gadgetry works in terms of maps to and from products. Here then is a rather nice example to meet early to motivate categorial ways of thinking. But not in this book.

Still, look at this for what it is rather than for what it purports to be. In other words,  look at this as a  set of detailed lecture notes which someone could use as back-up reading for perhaps the first half of a hard-core course, to keep things on track by checking/reinforcing definitions and key ideas, with added exercises  (notes which could then later be useful for revision purposes). Then Roman’s book does seem to be  pretty clearly done  and likely to be useful for some students. But if you were wondering what the categorial fuss is about and wanted an introductory book to draw you in, I doubt that this is it.

[Two grumbles. The book is pretty pricey for its length. And why, oh why, in an otherwise nicely produced paperback have the category theory diagrams been drawn in such an ugly way, given the available elegant standard LaTeX packages?]

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