The two Parts of my notes on category theory are now two separate PDFs, and all the chapters in Part I have now been revised (with the last chapters significantly rewritten since the last update). So here they are!
Part I is still very much work in progress, but there are more than enough major improvements since the last version to make it worthwhile to post this update. I think the structure of Part I, the overall order in which topics are tackled, should remain stable from now on: but I’ll certainly want to return to improve the treatment of some of those topics. So all comments on the current version will be immensely useful.
I’m inspired by some recent discoveries to re-start posting weekly links to online music videos. As before, they might be old or new, perhaps just part of a concert, perhaps just half an hour more or less. Let’s see how it goes again.
This week one of the most sheerly enjoyable videos yet — the wonderful Janine Jansen with seven other mostly young musicians playing the Mendelssohn Octet. This is about the best performance I have ever heard, played with such verve and such shared delight in the music that it is very affecting. [33 mins]
Oh, I know, it is such a cliché to mention it. But it is somehow difficult to get over the astounding fact that Mendelssohn was 16 when he wrote the Octet.
This may by now be old news, but there is a recently started journal Annals of Mathematics and Philosophy. It aims to encourage dialogue between contemporary mathematics and philosophy. Issues are (it seems) to be freely available online. The first issue is out at the journal’s website here.
I confess I didn’t find this issue an encouraging start, being mostly low-level stuff that wouldn’t have got near publication in e.g. Philosophia Mathematica. Except that there is a substantial, characteristically lucid, and very interesting piece by Timothy Gowers, “What makes mathematicians believe unproved mathematical statements?” which I much enjoyed reading.
I have added a short new chapter to Category Theory: Notes towards a gentle introduction. Just four-and-a-bit pages on “power objects” (i.e. objects which behave in a certain way as a powerset does in the category Set). This is really just an exercise in making the definition look tolerably well-motivated. I don’t want/need to run very far with the idea: I just want to have it in play for when I get round to first introducing the notion of a topos.
So the end of Part I of the Notes is in sight. I need to have a chapter expanding the current half-chapter on natural number objects, and another chapter expanding the current half-chapter on group objects. Then the Part will finish with a new chapter introducing the notion of a topos (all the ingredients are in place) and looking ahead, motivating the transition to Part II. As all my predictions about how long things will take turn out to be useless, I won’t venture one. But whenever I do get to the end of Part I, the plan is to pause and work through it again making it as clear and unified in level and approach as I can, rather than rush on to Part II.
Post-covid chez Logic Matters continuous to be exceedingly tedious. Tests may come back negative, but the fatigue continues with only very slow improvement. Par for the course we are told. If we had the energy to be really annoyed, we’d be really annoyed …
I have not been exactly full of logical energy! But I’ve been up to a little light re-editing of the already-revised chapters of Category Theory: Notes towards a gentle introduction. I’ve repaired some typos and a few thinkos, slightly speeded up a few sections which added unnecessary complications, and even added a couple of minor theorems. The main changes are at the very outset in the pre-categorial chapter on groups, and in the first two chapters on products.
Some of the typos/thinkos in the previous version are annoying enough for me to want to put the current corrected notes online right away. So I have …
Otherwise covid chez Logic Matters wends its tedious way along what seems to be the now most common trajectory, where the main continuing effect is considerable fatigue outlasting mild-ish cold-like symptoms. Could be a lot worse. Back to music and other postings, I hope, next week.
I intended to return from Perugia full of renewed energy, imagination, and zest. It hasn’t quite worked out like that. We have Covid. So far, it’s largely a matter of massive fatigue. Let’s hope that’s the worst it gets. So perhaps not much here on logical matters for a week or two.
Six days away in Perugia. Where we’d never been before, despite the daily flights from Stansted Airport near Cambridge to the Umbrian airport very close to Perugia. But three weeks or so ago, Mrs Logic Matters saw a very warm review of an exhibition to mark the five hundredth anniversary of the death of Perugino, and that prompted us to get our act together and organize travel and a hotel. (I can’t but wonder, though, what St. Francis of Assisi makes of having an airport named after him …)
And the exhibition is indeed wonderful, more than worth the trip. It is quite beautifully presented, small enough not to be overwhelming, large enough both to give you a sense of Perugino’s mastery and very illuminatingly to put the artist in some context. The photo above is from the Perugino2023 website here (worth exploring). We very much enjoyed taking the exhibition slowly, a couple of times.
Perugia’s centro storico is rather stunning. And as is frequently the way in Italian cities, the ground-floor spaces in the old, often very old buildings, are used to accommodate shops and cafes in (mostly) such remarkably decorous and sympathetic ways. So the streetscape is a visual delight without being in the least a museum piece. The weather was still too chill for people to be sitting outside cafes late into the evening — Perugia is on the top of a hill, and catches the wind. But even so, there was a bustle to the ancient Corso Vannucci, and we found ourselves caught up in its early evening passeggiata.
Away from the very centre, the city starts to tumble down the sides of the hill along steep narrow roads and down alleys which turn into staircases. We sent a lot of time just wandering around. But then of course there are all the churches worth visiting. And not least there is the Galleria Nazionale dell’Umbria, where the Perugino exhibition was held. The main two-floor gallery in the Palazzo dei Priori had a complete renovation only last year, and the result is a triumph. The spaces are just beautiful and the collection includes some real masterpieces, wonderfully well displayed. Visit GNU’s website and look through the rooms to get an impression.
Since we were staying in a hotel rather than an apartment we had to eat out at restaurants every night. Tough, eh? And while in Cambridge you can eat mostly badly or very badly in a dozen different cuisines, in Perugia you get to eat well or very well in the one Umbrian style.
Well, I exaggerate. But only a little. And we had the best meal out in two or three years at Luce Ristorante — wonderful ambience and service, superb food and wine (imaginative without being over-the-top). We paid at most half what a comparable meal would cost in the handful of “fine dining” places in Cambridge.
And as for coffee … Italian cafés have mostly avoided the plague of “artisanal coffee” which makes for undrinkable brews unless drowned in milk in the English way. So the espresso still tastes as the good lord intended. Which is a relief.
I’m hardly exploring the more obscure musical corners in these weekly posts! But no matter. This week, it’s the ridiculously gifted Noa Wildschut engagingly enjoying herself playing Bach with great verve and musicality. A delight to watch as Mrs Logic Matters particularly agreed. Just what is needed after a very busy and rather stressful week!
Another update to Category Theory: Notes towards a gentle introduction. This time, I have rewritten and indeed doubled the length of the chapter on subobjects and subobject classifiers. Much improved, I hope! I’ve also added two or three needed results at the appropriate places earlier in the notes, so that the proofs in this newly revised chapter can be snappy, and corrected some random typos I happened to notice.
(I’ll be pausing on categories now for a few weeks, with a holiday break coming up.)
The very first post was back on March 9th, 2006 (and I have just restored the link there to a short talk on Gödel’s theorem for mathematicians which I’d quite forgotten about ). Since then there have been — rather astonishingly — 1630 posts.
Of course, many of these were of merely passing interest (if that). But over the years, there have been some blog posts, or series of posts, of perhaps less ephemeral interest. So I’ve decided to rework the webpage that used to be unhelpfully labelled ‘Varia’ to become more of an ‘Archive’. I plan over the coming weeks not only link to a few papers, book reviews, and other pieces, but also to ‘best bits’ from the blog. It will take a while to organize this; but I’ll hopefully enjoy finding out what earlier time-slices of me thought!
Heavens: I’ve just realized that if I start the clock when I got my first job, the blog has been going for about a third of my academic life. Doesn’t time fly when you are having fun.