Update on formal anafunctors

After some very helpful comments, I have managed to finish updating my paper on formal anafunctors (original release announcement post) and have now sent it back to the journal. The length increased by about 60%, as I had missed including the proof that the associator isomorphisms were natural (7 pages! Including the diagram in this post) and that middle-four interchange holds. The referee also pointed out that the covering maps don’t really form a subcanonical pretopology, but something a bit weaker, and this weaker notion is all I use. It wasn’t so much a matter of tweaking the definition, but recognising the weakness of the definition.

So here it is: The elementary construction of formal anafunctors, arXiv:1808.04552.

For amusement, two of the new diagrams…

Now I need to write the cospans paper

The implications!

“…p ⇒ q is actually equivalent to ¬(p ∧ ¬q), and undergraduate lecturers consistently get it wrong.” -Andrew J. Bromage (@deguerre)

😲 …. it makes so much more sense written like this! Why have I never been shown this before?? Link to thread:

Elementary solution to Painlevé III

Last semester, when setting a question for my first-year calculus class, I was trying to do something that couldn’t be solved by online tools like Wolfram Alpha or Symbolab. In particular, I wanted to get the students to practice verifying that a particular function solves a particular differential equation, but without being able to solve the DE.

It turns out that for special parameter values the third Painlevé transcendant, namely

y'' = \frac{1}{y}\left( (y')^2 - BA^2/3 \right) + \frac{1}{x}(By^2 - y'),

has an absolutely elementary solution. I really wanted to track this down to its source, and ended up finding this:

N. A. Lukashevich, “On the theory of Painlevé’s third equation”, Differ. Uravn., 3:11 (1967), 1913–1923 Math-Net.Ru

To verify y = A\sqrt[3]{z} is a solution to the given case of Painlevé III above requires no clever tricks, no special knowledge. So I tried to make this a question in my (online, open-book, do-at-home) exam…

Not the final question. The parameters A and B were meant to be randomised.

It turns out that, at least the way I tried to do it, Wolfram Alpha and Symbolab can’t solve Painlevé transcendents 🙂


A new class of philosophically-minded mathematician that I just learned from the logician Paul Levy: smallist.

CH is a statement of third-order arithmetic. It doesn’t quantify over the universe of sets. GCH, on the other hand, does. For smallists, who take a platonic view of PPN (powerset of powerset of the naturals) but not of the universe of sets, this is a big difference. (http://www.cs.nyu.edu/pipermail/fom/2016-October/020149.html)

I guess it means a mathematician who doesn’t necessarily want in their axiomatic system arbitrary powersets, rather just the few that are needed for ‘ordinary’ mathematics (say up to PPPN, which is plenty to deal with differential geometry, differential equations, functional analysis, number theory, algebraic geometry over number fields or rings of integers therein etc). I think he just invented the word 🙂 but I like it. For a categorically-minded person like me, this means I could work in a pretopos with just a few powersets posited.

(Originally posted to Google+ on 11 November 2016)

MathOverflow question seeking answer, offering bounty

Last year I asked a question titled Direct comparison zig-zag between cochain theories on MathOverflow, and it has still no good answer. I’ve offered a bounty, and no takers yet. The satisfaction is yours for the taking! To quote from the end of the question:

…my hope is to show that the class of cochain theories is connected, assuming one exists, and then exhibit one. This may well be singular cohomology, or it might be something else. In particular I want to remove from the proof of uniqueness the privileged position that any one construction has. The only caveat is that I won’t be able to use any super-sophisticated machinery as this is a first course in algebraic topology. I’m happy to have an outline of how to unwind a sophisticated proof.

Teaching Algebraic Topology at the Australian Mathematical Sciences Institute Summer School

I’m pleased to finally announce that I’m teaching an Algebraic Topology course for the 2021 AMSI summer school in January and February! It will be an adaptation of the course I taught last year, omitting the material on covering spaces and fundamental groups. Instead, I aim to cover simplicial sets as well as \Delta-sets, and cover the Eilenberg–Steenrod material in more detail.

More excitingly, I aim to stream the lectures on Twitch. I haven’t set all this up yet, but I will hopefully be practicing streaming some of my own work process, maybe even developing some of the course notes, before January.

Want to see the real secrets of set theory in YouTube format?

tl;dr Set theorist Asaf Karagila is looking for YouTubers to collaborate.

My colleague and sometime rival Asaf is a top notch young set theorist who works a lot on pushing the frontier of the method of forcing. At one point we were in competition to construct, without using large cardinal assumptions, the first model of set theory where the axiom WISC failed. Asaf won, but as I was working in a different formalism, I still had the satisfaction of arriving at my own solution. This was right at the start of his academic career, and he’s only gone from strength to strength, recently being awarded a prestigious UK Future Leaders Fellowship.

The upshot is, he included explicitly in his Fellowship application that he would produce outreach videos about set theory, and is looking to collaborate with YouTubers with wide reach to achieve this. As he writes:

“There is a clear lack of good videos addressing set theoretic ideas, which I honestly believe that I can make at least somewhat accessible. And hopefully this will make set theory more accessible to the public, or at the very least, to other people interested in mathematics.”

He has set up a contact email if you are a YouTuber:

At the moment I’ve set up an email address, youtube2020@karagila.org, where you can email me. Let me know about your channel, what kind of content you want to make, etc. I cannot make any promises about money, but I’m always happy to advise with regards to content, should the need ever arise.

And if you are not a YouTuber, but want to see some more nitty gritty about what it is that set theorists do nowadays, point them to Asaf’s blog post! Asaf tells me that Numberphile and Tibees have already made contact, but if you are super keen to support the idea, it would help if viewers promoted the idea.

Now, if YouTubers want to make videos about category theory, on the other hand, then, ahem, I don’t mind having a chat 🙂 But they should talk to Asaf first, I don’t want to intercept his efforts!