Notes on Category Theory v.6b

Illness (not so great) followed by fortnight’s holiday (really excellent) stopped work on category theory for a while. Very slowly getting back to it. But in the meantime, a number of people have very kindly been sending corrections to the last version of the Notes. I have also tinkered in minor ways, improving the last chapter in particular. There are just about enough changes to warrant another “maintenance upgrade”, making some of the needed repairs and improvements.

I hope to have a couple more chapters on adjoints ready for prime time later in the month — but finding a neat expository path through the material is a challenge, so don’t hold your breath!

The plan at the moment is for another five or six more chapters in total to round off Part I of these Notes, on basic category theory (I’m not sure yet whether I also need to say anything about monads for what is going to follow). And then — having got the bit between my teeth — I’d like to continue, by discussing some logic and set theory in a categorial way in Part II of the Notes. Promises, promises.

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Peter Cropper, 1945–2015

84f2f3d0-d060-4682-b176-c5f3e1e9a488-1020x612It is very sad to hear of the untimely death of Peter Cropper, for forty years the inspirational and charismatic leader of the Lindsay String Quartet.

When we lived in Sheffield, we often went to hear the Lindsays play at the Sheffield Crucible Theatre (not the big hall, but the very intimate studio theatre where the players sit in  a central space surrounded by tiered seats, a few hundred people just yards away). These could be extraordinary occasions, which always involved Peter Cropper talking to the audience about what the Quartet was about to play. His enthusiasm and passionate involvement made for memorable evenings.  The playing wasn’t always immaculate — but “Who wants perfection? Perfection is sterile. We’re human beings.” Peter Cropper was also instrumental in setting up quite exceptional series of concerts over the years in  Sheffield’s ‘Music in the Round’ (I guess using his personal warmth and contacts to entice world-renowned musicians to this small venue).

The Lindsays were at the height of their musical achievement in the period when we were able to hear them so often, and the CDs that came out at this time — often, like the ‘Bohemians’ series, after series of performances in Sheffield and the Wigmore Hall — are consistently wonderful and are among the very best quartet performances we have. Peter Cropper’s playing on their Haydn disks shows a warmth and a delight in Haydn’s endless invention that is absolutely captivating. The quartet’s Beethoven is unsurpassed.

But I also have more personal memories. When at school, The Daughter was taught the violin by Nina Martin, Peter Cropper’s wife. And Nina would a couple of times a year arrange concerts of her pupils (and pupils of local viola and cello teachers, so quartets and baroque concertos could be played). After the concerts, some pupils and parents would often go back to Peter and Nina’s house, and crammed into the sitting room there would be more music. I can remember now a scratch, sight-read, performance of the first half of Mendelssohn’s Octet with six assorted school kids, including The Daughter on fourth violin, plus Nina busking on second viola, and Peter attacking the first violin part with his usual passion, and inspiring  a moment of magic from his impromptu ensemble.

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Logical snippets (again)

[Updating a post from fifteen months back.] For almost three years now, I’ve been a contributor to the useful question-and-answer site, math.stackexchange.com. This is a student-orientated forum, not to be confused with the truly wonderful mathoverflow.net which is its research-level counterpart. Think of my efforts as (hopefully) constructive procrastination on my part.

Of course, many of the questions on the site, including many I’ve found myself answering, are very ephemeral or very localized or based on very specific confusions. But a proportion of the exchanges to which I’ve contributed might, for one reason or another, be of some interest/use to some beginners and near beginners in logic.

So a while back, I put together a page of links to these logical scraps, morsels, excerpts, … snippets, shall we say. The links are grouped by level and/or topic.

This snippets page isn’t the most visited area of Logic Matters, but it gets more than enough hits to make it worth keeping alive. So — although I’ve been contributing to math.stackexchange rather less of late — I’ve just added links to some more recent postings there. Spread the word to students as/when appropriate.

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Postcard from Cornwall

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We have been in Cornwall for a fortnight. Saint Mawes, since you asked. Much to be recommended for riposo totale. We have already booked to return to the same place next year.

Now I’m back, this blog will splutter into life again. Though I’ve just been deleting a little rather than adding. I had been posting initial discussions of the opening chapters of John Burgess’s Rigor and Structure. I was, however, beginning to find the book surprisingly thin and unhelpful, and didn’t have anything useful to say: so rather than continue carping I’ve decided to remove those posts. I did read on further, to get to what were advertised as the main novel claims. But I must be missing the point as what I found seemed banal. I seem to be too out of sympathy with Burgess style and approach. Your mileage may, of course, very well vary.

For a sharply contrasting book, at least in the level of depth and care, can I instead recommend (well, I’m only a couple of chapters in, but it is going terrifically) Ian Rumfitt’s The Boundary Stones of Thought (OUP). This is a predictably serious discussion of the nature of logic in which Rumfitt defends classical logic against a variety of broadly anti-realist attacks. Exemplary and inspiring stuff.

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Notes on Category Theory v.6a

A number of people have very kindly sent corrections to the last version of Notes on Category Theory. There were some possibly confusing typos and also a downright wrong proof. Embarrassing. So here’s a “maintenance upgrade”, making some needed repairs.

[Added: a few more needed corrections are noted in the comments below.]

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The Open Logic Text

As you will very probably have already seen, The Open Logic Project (a team of serious and good people) has now made available an early public version of an open-source collaborative logic text, somewhat ploddingly called the Open Logic Text. 

There are two things to comment on here (eventually!), namely the Text itself — or at any rate, the current snapshot of an evolving text — and the open-source nature of the enterprise.

At a first quick glance, the Text does look rather uneven: there are 77 pages on first-order logic and beyond (some at quite an elementary level), 100 pages on computability, incompleteness, etc. (this looks like a solid graduate course), and then just 21 pages on sets (at a very much lower level of sophistication). Still, this is obviously exactly the sort of thing that should be covered in the Teach Yourself Logic Study Guide. So when I’ve had a chance to take a serious look, I’ll report back with my two pennies’ worth, maybe in a mid-year update to the Guide.

You can download the current version as a PDF. But as the Project site says of the Text,

… you can download the LaTeX code. It is open: you’re free to change it whichever way you like, and share your changes. It is collaborative: a team of people is working on it, using the GitHub platform, and we welcome contributions and feedback.

I will be really interested to see how this pans out in practice. Using GitHub is a notch or three above my current nerdiness grade. But I simply don’t know if this is me just not keeping up with everyone — or whether it is pretty typical for logicians to know a smidgeon of very basic LaTeX, with that being about their geek limit. Maybe, at least as a bit of exercise to keep the brain from entirely rusting up, I should take a look at this GitHub malarky about which I’ve heard tell before (any useful pointers to an idiot’s guide?). Then I could also report back about how the collaborative aspect looks to a complete beginner. Again, watch this space.

Posted in Books, Geek stuff, Logic | 11 Comments

Notes on Category Theory v.6

Here is an updated version of my on-going Notes on Category Theory, now over 150 pp. long. I have added three new chapters, at last getting round to the high point of any introduction to category theory, i.e. the discussion of adjunctions. Most people seem to just dive in, things can get a bit hairy rather quickly, and only later do they mention, more or less in passing, the simpler special case of Galois connections between posets (which transmute into adjunctions between poset categories). If there’s some novelty in the Notes at this point, it’s in doing things the other way around. We first have a couple of chapters on Galois connections — one defining and illustrating this simple idea, the other discussing a special case of interest to the logic-minded. Only then do we get round to generalizing in a rather natural way. We then find e.g. that two equivalent standard definitions of Galois connections generalize to two standard definitions of adjunctions (presented without that background, it isn’t at all so predicatable that the definitions of adjunctions should come to the same).  I do think this way in to the material is pretty helpful: I’ll be interested, eventually, in knowing how readers find it.

So we this is what we now cover:

  1. Categories defined
  2. Duality, kinds of arrows (epics, monics, isomorphisms …)
  3. Functors
  4. More about functors and categories
  5. Natural transformations (with rather more than usual on the motivation)
  6. Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
  7. Categories of categories: issues of size
  8. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
  9. An aside on Cayley’s Theorem
  10. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
  11. Representables (definitions, examples, universal elements, the category of elements).
  12. First examples of limits (terminal objects, products, equalizers and their duals)
  13. Limits and colimits defined (cones, limit cones: pullbacks etc.)
  14. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
  15. Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)
  16. [NEW] Galois connections (warming up for the general discussion of adjoint functors by looking at a special case, functions that form a Galois connection)
  17. [NEW] An aside for logicians, concerning a well-known example, the Galois connection between syntax and semantics.
  18. [NEW] Adjoints introduced. [Two different definitions of adjoint functors, generalizing two different definitions of Galois connections; some examples of adjunctions; a proof that the two definitions are equivalent.]

There will certainly be a few more chapters on adjoints. But don’t hold your breath, with a family holiday coming up and some other commitments. I haven’t decided yet whether eventually to add a chapter or two on monads (for monads seem a standard next topic to cover — e.g. in the last main segment of the Part III Tripos category theory course this year, the last chapter of Awodey’s book). Watch this space.

Posted in Category theory | 10 Comments

Barnaby Sheppard’s The Logic of Infinity — website

I haven’t yet reviewed Barnaby Sheppard’s The Logic of Infinity (CUP 2104) here — and I don’t know if I will, for even if time may be infinite, that allotted to me certainly isn’t! But when I dipped into the book, it looked a Really Good Thing which could of be real use to its intended audience of near beginners in mathematics whose imaginations might be captured by foundational questions.

Now I know only too well what it is like to publish a book with technical aspects and then find the inevitable typos and thinkos and sheer mistakes. At least the internet makes it possible to ease the pain a bit by giving you a second chance to explain what, really, you meant to say. But readers need to know where to look. So as a friendly gesture to a fellow author, let help me spread the word that Barnaby Sheppard has now set up a small website for errata for his book.

Posted in Books, Logic | 1 Comment

Brilliant indeed

51OEUHk5YiLI was going to post about the delights of Amsterdam as a place to visit for a week — the cityscapes, the cafes, the restaurants, the museums large and small, the whole urban experience, all even better than we had hoped. But more or less as soon as we got back, I was felled by a nasty attack of a recurrent problem, about which all I will say is thank heavens for penicillin. Though industrial quantities of antibiotics do leave you feeling still pretty flattened, so it has been a few days of staggering from bed to sofa and back. But as I begin to feel more human I’ve had plenty of time to finish the book I’d just started before going away. Like Amsterdam, this too has been lauded to the skies by those who know it. It has been a delight, in both cases, to find that other people’s really warm recommendations are more than deserved (it doesn’t always happen!). And since you certainly don’t need me to tell you more about Amsterdam, but you might not have heard of Elena Ferrante’s My Brilliant Friend — I hadn’t until a couple of months ago, from the much better read Mrs Logic Matters — maybe I’ll just sing its praises instead.

It really is absolutely wonderful. But I’m not going even to try, in my limping way, to say why. Rather let me point you to this New York Review of Books review of Ferrante’s oeuvre by Rachel Donadio, and/or this review from the New Yorker by James Wood. If these don’t get you reading, nothing will!

Posted in Books, Italian matters | 3 Comments

Notes on Category Theory v.5

Here is an updated version of my on-going Notes on Category Theory, now 130 pp. long. I have done an amount of revision/clarification of earlier chapters, and added two new chapters — inserting a new Ch. 7 on categories of categories and issues of size (which much expands and improves some briefer remarks in earlier versions), and adding at the end Ch. 15 saying something about how functors can interact with limits. There’s quite a bit more that could be said in this last chapter, and I’ll have to decide in due course whether to expand the chapter, or return to the additional topics later, or indeed to only mention some of those topics in the end (I’m trying to keep things at a modestly introductory level). But for the moment I’ll leave things like this and move on to a block of chapters on adjoints and adjunctions. So here’s where we’ve got to:

  1. Categories defined
  2. Duality, kinds of arrows (epics, monics, isomorphisms …)
  3. Functors
  4. More about functors and categories
  5. Natural transformations (with rather more than usual on the motivation)
  6. Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
  7. [New] Categories of categories: issues of size
  8. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
  9. An aside on Cayley’s Theorem
  10. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
  11. Representables (definitions, examples, universal elements, the category of elements).
  12. First examples of limits (terminal objects, products, equalizers and their duals)
  13. Limits and colimits defined (cones, limit cones: pullbacks etc.)
  14. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
  15. [New] Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)

Don’t hold your breath for the chapters on adjoints, though. After a very busy time for various reasons, I’ve a couple of family holidays coming up!

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