Category theory

Introducing Category Theory — the paperback!

The headline news: there is now an inexpensive (but quite acceptably produced) paperback of Introducing Category Theory. Amazon-only to minimize cost, ISBN 978-1916906396: US $14.99, UK £10.99, DE €14.82, IT €14.40, etc.

I’m very sure this could be improved in all kinds of ways. As I say at the end of the Preface, the current text is quite certainly not set in stone. Indeed, think of it as a ‘beta version’, functional though surely not bug-free. But I need to pause work on it for a while, so I thought I’d get a paperback version out for those who prefer to work from one. All corrections and suggestions for improvement will continue to be very gratefully received.

The text of course remains freely downloadable as a PDF from the categories page. (The earlier paperback of Part I is, for the moment at least, withdrawn.)

Introducing Category Theory — a full draft, at last!

There is now, at last, a full draft of Introducing Category Theory. You can download the PDF here.

The second half still needs more proof-reading and needs indexing. But I don’t envisage adding significantly to the content. After all — rather crazily for a book I didn’t originally set out to write — it is already 450 large format, small print, pages. Which is surely enough for an elementary, limited-ambition, introduction.

I hope, as a certain author once put it, it will provide interested readers with a ladder they can throw away after they have  climbed up it, now primed to tackle some of the standard books by real category theorists. (Though, unlike that author, I certainly don’t intend that  “anyone who understands me eventually recognizes [what I wrote] as nonsensical.”! Any nonsensical bits are plain mistakes.)

What’s the plan from here on? I want to complete the indexing pretty speedily, and do another proof-reading for the second half (though I seem to be increasingly bad at that!). And then I’m minded to promptly paperback it, though as a frankly acknowledged “beta version” with the expectation that I’ll certainly need to update it to correct typos and thinkos. But many readers will much prefer to work at least in part from a printed copy.

(I would have quite liked to have modest colour printing for some of the diagrams, to set off panels for theorems, etc. but that would more than double the book cost. The zero-royalties price for black-and-white Amazon print-on-demand, as with the other Big Red Logic Books, should be a tolerable £10.75, $14.90, €13.75 — still just the price of a few coffees.)

The shape of categorial things to come

The headline news: there is now a new version of my category theory notes, in one document, as Introducing Category Theory, available here.

In more detail: I’ve decided against my previous plan of two volumes, in part because there just wasn’t a satisfyingly natural way of carving things between them. (Friends and relations, by a large majority, also voted for one big book rather than two smaller ones.)

As the notes are now arranged, Part I says something about what can be found inside individual categories (products, quotients, limits more generally, exponentials and the like). Part II says a very little about those categories which are elementary toposes. Part III, which can be read independently of Part II, introduces the distinctive categorial  ideas of functors, natural transformations, the Yoneda Lemma, and adjunctions.

If you haven’t seen earlier versions, I have gone for a fairly conventional mode of presentation but at a pretty gentle pace (which makes for quite a long book — but I make no apology for that: faster-track alternatives are available in you want them!). The result is aimed at those who want an entry-level warm-up before taking on an industrial-strength graduate-level course, or perhaps just want to get an idea of that the categorial fuss is about.

The state of play? Part I should be in a good state. One chapter is missing from Part II. Part III needs a concluding note on further reading. The index needs work. But I’m getting there!

A print-on-demand version will be inexpensively available (Amazon only, to minimize cost) in due course — by the end of July, I hope.

A print-on-demand version of Part I (and a little more) has been available under the title Category Theory I: A Gentle Prologue. This now strikes me as making an unsatisfactory division of the material, stopping precipitously. [Updated] However, I have decided to keep it in print for the moment, and in a day or so a new version, updated to correct typos and a few thinkos, should go live on Amazon.

Two category theory books

One of the best freely available sets of lecture notes on category theory available is the one by Paulo Perrone, which can be downloaded from the arXiv here. He has now turned these notes into a printed book, Starting Category Theory from World Scientific. There is a long added chapter on monoidal categories, but the earlier five chapters appear again with only some small additions as far as I can see on a quick browse (the significantly higher page count for these chapters being mostly due to the difference in formatting).

Since the lecture-note version is freely and officially available, I won’t comment in detail here: suffice to say that Perrone makes the material pretty accessible, and I can well imagine many students liking his style. I really, really doubt that they (or you) will want to fork out for the book, however. Its USA price is $148 (there’s a whopping $11 discount on Amazon); its UK price is £135. That is worse than absurd.

I may just possibly have said something here about another category theory book’s being on its way. Namely, my Category Theory II. But plans are a bit fluid right now. For various reasons, I’m increasingly minded to in fact combine Parts I and II (with again a slightly re-arranged ordering of chapters) into a single big book, perhaps reverting to that earlier title Category Theory: Notes towards a gentle introduction for the whole.

In fact the cost of one big printed book would only be about the price of a large coffee more than a separate Category Theory II, and the small additional overall cost to someone who already has Category Theory I would be more than compensated by avoiding the repeated annoyance of having to chase up cross-references between books (or between PDFs). And there are other significant potential gains too: the combined book as I now see it has a nicer shape. But I’ll keep thinking this through: so watch this space.

Category Theory I — now in paperback

Sound the trumpets! Or at least, let me give a small toot …

Category Theory I: A gentle prologue  is now available as a print-on-demand paperback. This is Amazon only (sorry, but that’s easiest for me and cheapest for you), with ISBN 1916906389, at £5.99, $8.25, €7.50. Those prices are only trivially rounded up from the minimum possible (e.g. from £5.90), so I’m not taking any significant royalties.

I’m thinking of the paperback as still a beta version of the text. Functional, and I hope with no horrible mistakes, but surely not bug-free. I’ll still be very happy, then, to get corrections and comments and suggestions for improvement. The Amazon print-on-demand system makes future updates of the file for printing very straightforward and cost free.

The PDF is of course still freely downloadable from the category theory page; but many prefer to work from a printed copy. So, since the paperback is the price of — what? — just three coffees, and is actually quite nicely produced, why not treat yourself!

Ana Agore, A First Course in Category Theory

The most recommended introductory books on category theory (at least for pure mathematicians) are probably those by Steve Awodey, Tom Leinster, and Emily Riehl. All three have very considerable virtues. But for differing reasons, each presents quite steep challenges to the beginner (especially for self-study). Having, back in the day, worked through Awodey’s book with a reading-group of super-smart Cambridge Part III (i.e. graduate) students, I can only report that we found it engaging but a much bumpier ride than the author surely intended. Leinster’s shorter book, although my favourite, is often quite compressed and I’m told that students can again find it quite tough for that reason. Riehl’s book is full of good things — her title Category Theory in Context points up that she is particularly seeking to make multiple connections across mathematics. But she goes at pace and the connections made can be distractingly/dauntingly sophisticated.

So there is certainly room on the shelf for another introductory book, especially one advertised as being “unlike traditional category theory books, which can often be overwhelming for beginners. …[It] has been carefully crafted to offer a clear and concise introduction to the subject. … the book is perfectly suited for classroom use in a first introductory course in category theory. Its clear and concise style, coupled with its detailed coverage of key concepts, makes it equally suited for self-study.” So: does Ana Agore’s recently published A First Course in Category Theory (Springer, Dec. 2023) live up to the blurb?

Here’s the very first sentence of Chapter 1: “We start by setting very briefly the set theory model that will be assumed to hold throughout.” Which is garbled English. Quite unsurprising, I’m afraid, from Springer who don’t seem to proof-read their books properly these days. And I do wonder whether Agora has run her text past enough readers including a native speaker or two. For in fact there are quite a few unEnglish sentences. Fortunately, the intended message is only occasionally obscure, at least to this reader who has the advantage of knowing what Agore should be saying. I suspect, however, that some — especially if English is not their first language — may sometimes stumble.

The Preface tells us that the book is based on lecture notes from a graduate course. And that’s how it reads. We get action-packed notes, with a lot of detail given at a relentless pace, and with really very little added motivating classroom chat. The typical approach is to plonk on the table a categorial definition without preliminary scene-setting, and then give a long (sometimes very long) list of examples. And the level of discussion sometimes seems rather misplaced — is it really helpful for the introduction of categorial ideas to be interrupted, as early as p. 7, by an unobvious argument more than a page long to show that epimorphisms in Grp are surjective?

Again as early as p. 12, we are given the categorial definition of a subobject of C as an equivalence class of monics with codomain C. What could motivate pulling that strange-seeming rabbit out of the hat? We aren’t told. Rather, we quickly find ourselves in a discussion of how the definition applies in KHaus vs Top.

Another case: on p. 24 the definition of a functor is served up ‘cold’, followed by thirty-five examples. Or more accurately, we get thirty-five numbered items, but general points (e.g. that functors compose) are jumbled in with particular examples.

All in all, this does read rather like handout-style notes expanded with more proofs written out and with multiple extra examples, but without the connecting tissue of classroom remarks which can give life and direction to it all and which the self-studying reader is surely going to miss rather badly.

What does the book cover? How is it structured?

There are three long chapters. Chapter 1 (82 pp.) is on Categories and Functors, taking us up to the Yoneda Lemma. Chapter 2 (70 pp.) is on Limits and Colimits. Chapter 3 (98 pp.) is on Adjoint Functors. There follows a welcome chapter (26 pp.) of solutions to selected exercises.

But note that although Agore tells us about subobjects early on, we don’t get round to subobject classifiers. We meet limits and colimits galore, but we don’t meet exponentials. And again as contrasted with e.g. Awodey, while of course we get to know about categories of groups and groups as categories, we don’t get to know about groups in categories, internal groups.

In a little more detail, Chapter 1 covers what you would expect, basic definitions and examples of categories, types of arrows and special objects (like initial/terminal objects), functors, natural isomorphisms and natural transformations more generally, hom-functors and representables, ending up with Yoneda. There are some oddities along the way — the idea of elements as arrows from 1 (like the idea of ‘generalized elements) is never mentioned, I think, while the idea of a universal property makes its first appearance on p. 16 but seems never to be given a categorial treatment.

Tom Leinster has written “The level of abstraction in the Yoneda Lemma means that many people find it quite bewildering.” It’s a good test for an introductory book how clear it makes the lemma (in its various forms) and now natural the relevant proofs seem. How does Agore do? Here’s her initial statement.

She then adds that the bijections here, for a start, form a natural transformation in C:

If you are reading this review you are quite likely to know what’s going on. But if you were quite new to the material, I bet — for a start — that these notational choices won’t be maximally helpful, and the ensuing pages of proofs will look significantly messier and harder work than they need to be. So I certainly wouldn’t recommend Agore’s pages 70-77 as my go-to presentation of Yoneda.

Chapter 2 on limits and colimits continues in the same style. So the first definition is of multiproducts (rather than softening us up with binary products first). There’s no initial motivation given: the definition is stated and some theorems proved before we get round to seeing examples of how the definition works out in practice in various categories. We then meet equalizers and pullbacks done in much the same spirit (I don’t suppose anyone will be led astray, by the way, but contrary to her initial definition of a commuting diagram, Agore now starts allowing fork diagrams with non-equal parallel arrows to count as commuting).

On the positive side, I do very much approve of the approach of first talking about limits over diagrams, where a diagram is initially thought of as a graph living in a category, before getting fancy and re-conceptualizing limits as being limits for functors. And if you have already met this material in a less action-packed presentation, this chapter would make useful consolidating material. But, I’d say, don’t start here.

And much the same goes for Chapter 3 on adjunctions, which gets as far as Freyd’s Adjoint Functor Theorem and the Special Adjoint Functor Theorem. This is another rather relentless chapter, but with more than the usual range of examples. Some proofs, such as the proof of RAPL, seem more opaque than they need to be. Again, I wouldn’t recommend anyone starting here: but treated as further reading it could be a useful exercise to work through (depending on your interests and preferred mathematical style).

So the take-home verdict? The book advertises itself as a ‘first course’ and as suitable for self-study. However, I do find it pretty difficult to believe it would work well as both. Yes, I can imagine a long graduate lecture course, with this book on the reading list, as potentially useful back-up reading once the key ideas have been introduced in a more friendly way, with more motivating classroom chat. But for a first encounter with category theory, flying solo? Not so much.

Another categorial update

It’s been a month since I last posted about the category theory project, so a quick update  — and the end really is in sight!

  • I’ve just put online another revised version of Category Theory I. Little has changed except for some more corrections of typos (with particular thanks to Georg Meyer) and a few small changes for added clarity (with particular thanks to John Zajac). I’ve also made a few very small changes to better fit with what happens in Category Theory II as I steadily revise that.
  • More significantly, there is another version of Category Theory II linked on the category theory page. The old chapters on adjunctions are now in a much better state. I don’t think I found any horrendous errors, but the story is (I certainly hope!) a lot clearer in a number of key places.
  • In fact, the bit of recent work on this that I’m most pleased with is probably the proof of ‘RAPL’ (Right Adoints Preserve Limits). Tom Leinster and Steve Awodey offer fancy-but-unilluminating proofs. I spell out the sort of bread-and-butter proof idea I got from Peter Johnstone’s lectures (and my version is perhaps a little clearer for a first encounter than Emily Riehl’s?).
  • It’s a judgement call where to stop. For example, I still reckon (as I did before) that the Adjoint Functor Theorems are just over the boundary, as far as what is really appropriate for an entry-level introduction. However, I do now say just a very little about monads (so at least you know what the idea is), though I might yet add another example or two.
  • I still need to revise the last three chapters of Category Theory II. They should be in a reasonable basic state as these are the same final chapters that — in the previous arrangement — appeared as the last chapters in the 2023 published version of Category Theory I. However, in the somewhat more advanced context of Category Theory II it might be appropriate to expand the discussions a bit.

I’d hoped that Category Theory II would be paperback-ready by the end of this month. There have been unforeseen distractions. But I’m not far off. Watch this space.

A categorial update

Some categorial news:

  • I have just withdrawn Category Theory I: Notes towards a gentle introduction from sale as a pbk. There is going to be a new pbk edition, with a slightly different title, shortly — and now that plans are firmly under way, I don’t want anyone splashing out their hard-won pennies today only to find that a shiny new update is available a few weeks later.
  • You can download a draft of the new Category Theory I: A gentle prologue from the category theory page here. The obvious major change is that the chapters on elementary toposes at the end of the previous version have been moved to Part II, and a few initial chapters on functors moved from Part II into Part I. One result is that all of Category Theory I can indeed be thought of as a gentle prologue to some core topics in category theory.
  • The current draft of Category Theory II can also be downloaded. It has the new subtitle Four basic themes with groups of chapters covering (A) more on functors, including natural transformations, (B) around and about Yoneda, (C) adjunctions (D) a little on elementary toposes. I hope that a pbk version will be done and dusted by Easter.
  • I would still hugely welcome comments and corrections on both Parts. And indeed, even when paperbacked for those who like me prefer working from a printed copy, I’ll continue to think of them as beta versions — largely functional and I hope not too buggy but still work in progress.

It’s been sort-of enjoyable trying to get this stuff straighter in my mind, and a few friendly souls have told me that they’ve found my efforts helpful. But it really is (past) time I got back to other logic matters ….

Big Red Logic Books: 2024 plans

If you are new here, then here is the default page about the Big Red Logic Books

As I’ve noted before, self-publishing seemed exactly appropriate for the Big Red Logic Books. They are aimed at students, so why not make them available as widely as can be? — free to download as PDFs, for those happy to work from their screens, and at minimal-cost as print-on-demand paperbacks for the significant number who prefer to work from a physical copy. I posted reports of how things went in 2021 and 2022, half-hoping to encourage a few others to adopt the same sort of publishing model (though of course recognizing that those in early or mid career need the status points that come from conventional book publication). And I offered to give advice on the nuts and bolts of self-publishing to anyone interested. But response came there none. So I won’t bother to give a detailed report for sales and downloads in 2023. Rather, here are just a few headlines, and some thoughts about what comes next. Taking the books in the order of first publication on Logic Matters:

An Introduction to Gödel’s Theorem (2020: corrected reprint of CUP 2nd edition of 2013). Sales and downloads in 2023 slightly down on 2022 — but still almost 600 paperbacks sold in the year. I’m inclined to leave well alone, as many readers like the book as it is! (No, I’m not making a fortune! — the paperback prices are set so that total royalties are now zero for some books and pennies for others, together approximately covering the cost of keeping Logic Matters online.)

An Introduction to Formal Logic (2020: corrected reprint of CUP 2nd edition). Sales up over 20% at over 1500, downloads up over 55% compared with the previous year. Perhaps two or three more lecturers are using it as a course text. The absolute figures aren’t great, but then there are so many other intros to logic to choose from. There’s part of me that would like to one day write a third edition, or rather write a somewhat different Another Introduction … But whatever happens, I’ll leave this version available and in print, as it would be so annoying for those who have adopted the text if I dropped it!

Gödel Without (Too Many) Tears (2021, and then a second edition in late 2022). I thought that this much shorter book would for many be much preferred to IGT. However, after initially high sales for GWT, there now seems to be a steady pattern of the bigger book having 50% more sales and downloads. Unexpected, but I’m happy for IGT to be doing so well.

Beginning Mathematical Logic (2022) This descendant of the Teach Yourself Logic Study Guide is by far the most downloaded of the books. But it also sold well over 600 copies in paperback in 2023, to my genuine surprise. A considerable success then — but I suppose it is a text without obvious competitors.

Category Theory I (2023) New in August, and monthly sales and downloads already comparable to those of IGT. Again a cheering surprise since I have no standing on this topic, and it is only half a book — where, you might ask, is a finished second part?

So that’s the state of play at the turn of the year. What comes next? Obviously I need to finish the promised Category Theory II. But in fact I’ve changed my mind about what should go in Part I and what in Part II, pulling some chapters on functors into Part I, and moving the elementary discussion of toposes into Part II. The new edition of Category Theory I is on my desk as I write this, waiting to be proof-read. And I hope Part II will be print-ready by the end of February, though I’ll continue posting drafts as I go along.

I then want to return to BML, which needs an end-to-end rewrite (perhaps particularly on first-order logic where I want to rethink my recommendations). But that is going to take some time — a new edition of Beginning Mathematical Logic in 2025, Deo volente? But in the meantime, I ought quickly to do a revised reprint at least to correct a lot of known typos, and to add a page about some books published since early 2022.

That should all keep the grey cells ticking over. Watch this space …

Not Florence …

In latter years, Before Covid, we went to Florence a number of times before Christmas. It is a real delight then, when the city is largely free of other tourists. But it was not to be, this year. So there is only my virtual self, slouching down a deserted backstreet, as conjured up by ChatGPT.

I’ve been writing a bit about categories instead, which is distracting but hardly compensates. But it does mean that now all but the current last chapter of Category Theory II has been updated. It has taken me longer than it should have done, but the newly revised penultimate chapter is (I hope) both tidier and more accurate than it was. You can download the whole current draft here. And who knows? — with just one chapter to go, there might be a complete revised draft by the end of the year. Then I’ll have to think exactly what I’m going to do with it! (As always, you can download Category Theory I and other categorial goodies here.)

Later: there is now an updated first pass through the last chapter too …

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