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November 15, 2024
Axiomatic Set Theory 9: The Axiom of Choice
Posted by Tom Leinster
Previously: Part 8.
It’s the penultimate week of the course, and up until now we’ve abstained from using the axiom of choice. But this week we gorged on it.
We proved that all the usual things are equivalent to the axiom of choice: Zorn’s lemma, the well ordering principle, cardinal comparability (given two sets, one must inject into the other), and the souped-up version of cardinal comparability that compares not just two sets but an arbitrary collection of them: for any nonempty family of sets , there is some that injects into all the others.
The section I most enjoyed writing and teaching was the last one, on unnecessary uses of the axiom of choice. I’m grateful to Todd Trimble for explaining to me years ago how to systematically remove dependence on choice from arguments in basic general topology. (For some reason, it’s very tempting in that subject to use choice unnecessarily.) I talk about this at the very end of the chapter.
November 8, 2024
Axiomatic Set Theory 8: Well Ordered Sets
Posted by Tom Leinster
Previously: Part 7. Next: Part 9.
By this point in the course, we’ve finished the delicate work of assembling all the customary set-theoretic apparatus from the axioms, and we’ve started proving major theorems. This week, we met well ordered sets and developed all the theory we’ll need. The main results were:
every family of well ordered sets has a least member — informally, “the well ordered sets are well ordered”;
the Hartogs theorem: for every set , there’s some well ordered set that doesn’t admit an injection into ;
a very close relative of Zorn’s lemma that, nevertheless, doesn’t require the axiom of choice: for every ordered set and function assigning an upper bound to each chain in , there’s some chain such that .
I also included an optional chatty section on the use of transfinite recursion to strip the isolated points from any subset of . Am I right in understanding that this is what got Cantor started on set theory in the first place?
November 5, 2024
The Icosahedron as a Thurston Polyhedron
Posted by John Baez
Thurston gave a concrete procedure to construct triangulations of the 2-sphere where 5 or 6 triangles meet at each vertex. How can you get the icosahedron using this procedure?
Gerard Westendorp has a real knack for geometry, and here is his answer.
November 2, 2024
Summer Research at the Topos Institute
Posted by John Baez
You can now apply for the 2025 Summer Research Associate program at the Topos Institute! This is a really good opportunity.
Details and instructions on how to apply are in the official announcement.
A few important points:
- The application deadline is January 17, 2025.
- The position is paid and in-person in Berkeley, California.
- The Topos Institute cannot sponsor visas at this time.
For a bit more, read on!
November 1, 2024
Axiomatic Set Theory 7: Number Systems
Posted by Tom Leinster
Previously: Part 6. Next: Part 8.
As the course continues, the axioms fade into the background. They rarely get mentioned these days. Much more often, the facts we’re leaning on are theorems that were deduced from theorems that were deduced — at several removes — from the axioms. And the course feels like it’s mostly converging with any other set theory course, just with the special feature that everything remains resolutely isomorphism-invariant.
This week we constructed , , and . This was the first time in the course that we used the natural numbers axiom, and that axiom did get cited explicitly (in the first few pages, anyway). We had to use the universal property of to define sums, products and powers in , and to prove the principle of induction.
I think my highlight of the week was a decategorification argument used to prove the classic laws of natural number arithmetic. Read on…
October 31, 2024
Adjoint School 2025
Posted by John Baez
Are you interested in using category-theoretic methods to tackle problems in topics like quantum computation, machine learning, numerical analysis or graph theory? Then you might like the Adjoint School! A lot of applied category theorists I know have gotten their start there. It can be a transformative experience, in part thanks to all the people you’ll meet.
You’ll work online on a research project with a mentor and a team of other students for several months. Then you’ll get together for several days at the end of May at the University of Florida, in Gainesville. Then comes the big annual conference on applied category theory, ACT2025.
You can apply here starting November 1st, 2024. The deadline to apply is December 1st.
For more details, including the list of mentors and their research projects, read on.
October 29, 2024
Triangulations of the Sphere (Part 2)
Posted by John Baez
Thurston’s paper Shapes of polyhedra and triangulations of the sphere is really remarkable. I’m writing about it in my next column for the Notices of the American Mathematical Society. Here’s a draft — which is also a much more detailed version of an earlier blog post here.
If you have suggestions or corrections, please let me know: I can try to take them into account before this is due on November 5th. Just don’t ask me to make this longer: I have a strict limit.
October 26, 2024
Axiomatic Set Theory 6: Gluing
Posted by Tom Leinster
Previously: Part 5. Next: Part 7.
A category theorist might imagine that a chapter with this title would be about constructing colimits, and they’d be half right.
Triangulations of the Sphere (Part 1)
Posted by John Baez
I’m writing a short column about this paper:
- W. P. Thurston, Shapes of polyhedra and triangulations of the sphere.
Let me describe one of the key ideas as simply as I can.
October 21, 2024
Octoberfest 2024
Posted by John Baez
The Octoberfest is a noble tradition in category theory: a low-key, friendly conference for researchers to share their work and thoughts. This year it’s on Saturday October 26th and Sunday October 27th.
It’s being run by Rick Blute out of the University of Ottawa. However, the meeting is entirely virtual, so you can join from anywhere using this zoom link. The talks will be recorded, and shared publicly later.
Below you can see a schedule. All times are Eastern Daylight Time (EDT).
I can’t resist mentioning that I’m giving a keynote talk on 2-rigs in topology and representation theory at 2 pm on Saturday. It’s about some work with Joe Moeller and Todd Trimble.
October 18, 2024
Axiomatic Set Theory 5: Relations
Posted by Tom Leinster
Previously: Part 4. Next: Part 6
I called this chapter of my course “Relations”, but I should have called it “Specifying subsets and functions”, because that’s what it’s all about. This week, we saw that it’s possible to define subsets of a set by expressions like
and functions by expressions like
I didn’t want to drown the students in notation, so I didn’t give precise definitions of “property” and “formula”. Instead, I aimed to give them practical tools that would apply to situations they’re likely to meet.
October 17, 2024
Associahedra in Quantum Field Theory
Posted by John Baez
I haven’t been carefully following quantum field theory these days, but some folks on the Category Theory Community Server asked me what I thought about recent work using the ‘amplitudohedron’ and other polytopes, so I decided to check out these videos:
- Nima Arkani-Hamed, Advanced class on amplitudes.
There are 5, and so far I’ve only finished watching the first. But I have to say: I enjoyed it more than any lecture on physics I’ve seen for a long time!
Arkani-Hamed has the amusing, informal yet clear manner of someone like Feynman or Coleman. And he explains, step by step, how imaginary particle physicists in some other universe could have invented the associahedra just by doing scattering experiments and looking for poles in the S-matrix. That blew my mind.
October 11, 2024
Axiomatic Set Theory 4: Subsets
Posted by Tom Leinster
Previously: Part 3. Next: Part 5
This phase of the course is all about building up the basic apparatus. We’ve stated our axioms, and it might seem like they’re not very powerful. It’s our job now to show that, in fact, they’re powerful enough to do just about everything with sets that mathematicians ever want. We began that job this week, with a chapter on subsets.
October 10, 2024
2-Rigs and the Splitting Principle
Posted by John Baez
We’re done!
- John Baez, Joe Moeller and Todd Trimble, 2-rig extensions and the splitting principle.
Our paper categorifies a famous method for studying vector bundles, called the ‘splitting principle’. But it also continues our work on representation theory using categorified rigs, called ‘2-rigs’. We conjecture a splitting principle for 2-rigs, and prove a version of it in the universal example.
But we also do more. I’ll only explain a bit, today.
October 4, 2024
Axiomatic Set Theory 3: The Axioms, Part Two
Posted by Tom Leinster
Previously: Part 2. Next: Part 4
This week, we finished formulating the axioms of the Elementary Theory of the Category of Sets.