Category theory

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  • I have just withdrawn Category Theory I: Notes towards a gentle introduction from sale as a pbk.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 …

Back at last to matters categorial

Well, all that took much, much longer than planned! Having spent days sorting the shared books round the house, I was on a roll, and found myself seriously tackling my study for the first time in years. In the end, perhaps seven or eight feet of books have gone. And I feel all the better for it. A lot of second-division philosophy I certainly won’t be looking at again; no more baleful stares from those books that I’ll now never get round to reading; a couple of feet of mathematics books has gone too, texts I surely won’t seriously look at again (and which anyway are all available as PDFs in the familiar place we don’t talk about). But that leaves a well-populated wall. And I’ve kept most of Logic Corner for now too. Because I have plans …

Of course, most of you will think that that is still a quite ridiculous number of books. But then I belong to that thinning generation of academics for whom (in many cases) buying books was all tied up with our identity in what I suppose now seems rather foolish ways.

That major sort-out was all very distracting, though very good to get done. But I’ve meanwhile been tinkering away on and off with Category Theory II. In fact, I’ve made quite a few little presentational and other changes in earlier chapters, as well as splitting the material on Yoneda into two chapters again, and then I have newly tidied the chapter on representables. I guess I still want to say a little about the more elementary implications of Yoneda, but enthusiasts can download the latest version here. That leaves just the four chapters on Galois connections and adjunctions for initial revision. Then we’ll have to see how things stand. But the way things are going, Parts I and II together will already be a little over 400 pages which is surely enough for what is supposed to be a Gentle Introduction (which I never really planned to write in the first place). So at the moment I’m inclined to do no more than tie up a loose end or two and then stop.

Yoneda Without (Too Many) Tears

The newly revised chapter from Category Theory II is now a draft stand-alone PDF, Yoneda Without (Too Many) Tears.

All comments and corrections welcome before I perhaps put a revised version on the arXiv. It took me some effort back in the day to get this clear enough in my own mind; and I have tidied up the presentation, perhaps some others might appreciate a helping hand through the proof!

How bewildering is the Yoneda embedding?

There’s now another one-chapter update for Category Theory II.

There’s some minor earlier tinkering, but Chapter 37 has been considerably revised. The proofs leading up to what I call the Restricted Yoneda lemma and the Yoneda Embedding Theorem have been tidied up, and should be much clearer. And the final section “Yoneda meets Cayley” — which was a mess, almost incoherently so — is now crisp and clear. I hope!

Tom Leinster has written “The level of abstraction in the Yoneda Lemma means that many people find it quite bewildering.” While Awodey calls it “the single most used result” of category theory. So: bewildering but centrally important?

Well, I really do hope the decaffeinated version of Yoneda in Chapter 37 really is plain sailing. There’s basically one small idea — you can use a \mathsc{C}-arrow f\colon B \to A very simply to construct a natural transformation between hom-functors \mathsc{C}(A, --) and \mathsc{C}(B, --) — and then all the rest is pretty much applying definitions in obvious ways. So far, I hope, not bewildering at all!

What to read?

What recent, new, and forthcoming logic/phil. maths  books have caught your attention?

This first little book, forthcoming in January from CUP in the very mixed quality Cambridge Elements series, looks promising. Maddy, at least, reliably writes interestingly and well (hopefully, she keeps her co-author from getting too mired in technicalities). And the topic is a great one. “This Element addresses the viability of categoricity arguments in philosophy by focusing with some care on the specific conclusions that a sampling of prominent figures have attempted to draw … It begins with Dedekind, Zermelo, and Kreisel, casting doubt on received readings of the latter two and highlighting the success of all three in achieving what are argued to be their actual goals. These earlier uses of categoricity arguments are then compared and contrasted with more recent work of Parsons and the co-authors Button and Walsh. …  the Element concludes that categoricity arguments have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.” So this is certainly on my list.

I recently looked at Justin Khoo’s  The Meaning of If  published last year by OUP. Anyone writing an intro logic book (and I still hanker after a third edition of IFL) wants — or ought to want — to have something sensible to say about the relation of  ‘if’ and ‘⊃’, though some do duck the task. So I’m always interested to see what people are writing these days on the topic. But I can’t say I got much out of this. One of the phenomena here is that, however ‘if’s work in the wider world, in mathematics regimenting them by a connective  ‘⊃’ governed by the usual rules (acceptable to classical logic and constructive logic alike) seems to work a treat, at least once we distinguish plain ‘if’s from the ‘imply’s we regiment using turnstiles. But there isn’t a word about this in Khoo’s book (you look in vain for anything about mathematics, or indeed about “conditional proof”, or “supposition”, and so on). So whatever the virtues of this book — which I confess didn’t impress me — it will be of no real interest to logicians.

Erik Stei’s Logical Pluralism and Logical Consequence was published early this year (at a disgraceful price) by CUP. From the blurb: “The logical pluralist challenges the philosophical orthodoxy that an argument is either deductively valid or invalid by claiming that there is more than one way for an argument to be valid. In this book, Erik Stei defends logical monism, provides a detailed analysis of different possible formulations of logical pluralism, and offers an original account of the plurality of correct logics that incorporates the benefits of both pluralist and monist approaches to logical consequence.” OK, that looks as if it should be just up my street, as the topic is basic and important and I’m all for calming down debates by trying to draw out what each side has got right. My first impressions, though, on reading early pages have so far not been that encouraging. But I’ll certainly try again, and let you know.

On my desk right now, though, is Introducing String Diagrams: The Art of Category Theory by Ralf Hinze and Dan Marsden, recently published by CUP. This is a comp. sci. book in origin, and it is taking me a while to get the measure of it. But the book comes much praised, so I shall press on in the hope of pennies starting to drop with satisfying clunks …

Meanwhile, I’ve revised a couple chapters of my own entry-level (and hyper-conservative?) Category Theory II. Both these chapters, one on categories of categories and one on functor categories, have been much revised and in places simplified, so I hope work much better. There remain two groups of chapters to revise, one group on the Yoneda lemma and related stuff, one group on adjunctions. Fun topics. I remain quite undecided, though, about how things will go after this initial round of revisions of old material.

Another Category Theory II update

There have been considerable distractions over the last month, but I hope to get back to a few more logical blog posts this month.

For those interested, there’s now another small update for Category Theory II, in which I bring some considerations of size to the beginning of Part II in a tidier way, and then later I make what I say at different points about (locally) small categories more consistent. I adopt, for reasons explained, an idea in Roy Crole’s Categories for Types that defines e.g. a small category not as have a set of objects but as having objects that are indexed by a set.  I’m also a bit clearer about the differences between different ways of defining categories (something that didn’t really affect the discussions in Part I). Slow progress, but this required some actual thought and not just juggling technicalities!

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