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December 4, 2024

ACT 2025

Posted by John Baez

The Eighth International Conference on Applied Category Theory (https://easychair.org/cfp/ACT2025) will take place at the University of Florida on June 2-6, 2025. The conference will be preceded by the Adjoint School on May 26-30, 2025.

This conference follows previous events at Oxford (2024, 2019), University of Maryland (2023), Strathclyde (2022), Cambridge (2021), MIT (2020), and Leiden (2019).

Applied category theory is important to a growing community of researchers who study computer science, logic, engineering, physics, biology, chemistry, social science, systems, linguistics and other subjects using category-theoretic tools. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, strengthen the applied category theory community, disseminate the latest results, and facilitate further development of the field.

If you want to give a talk, read on!

Posted at 2:41 AM UTC | Permalink | Post a Comment

November 22, 2024

Axiomatic Set Theory 10: Cardinal Arithmetic

Posted by Tom Leinster

Previously: Part 9.

The course is over! The grand finale was the theorem that

X×YX+Ymax(X,Y) X \times Y \cong X + Y \cong max(X, Y)

for all infinite sets XX and YY. Proving this required most of the concepts and results from the second half of the course: well ordered sets, the Cantor–Bernstein theorem, the Hartogs theorem, Zorn’s lemma, and so on.

I gave the merest hints of the world of cardinal arithmetic that lies beyond. If I’d had more time, I would have got into large sets (a.k.a. large cardinals), but the course was plenty long enough already.

Thanks very much to everyone who’s commented here so far, but thank you most of all to my students, who really taught me an enormous amount.

Part of the proof that an infinite set is isomorphic to its own square

Posted at 3:27 PM UTC | Permalink | Followups (5)

November 15, 2024

Axiomatic Set Theory 9: The Axiom of Choice

Posted by Tom Leinster

Previously: Part 8. Next: Part 10.

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 (X i) iI(X_i)_{i \in I}, there is some X iX_i 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.

Section of a surjection

Posted at 2:26 PM UTC | Permalink | Followups (24)

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 XX, there’s some well ordered set that doesn’t admit an injection into XX;

  • a very close relative of Zorn’s lemma that, nevertheless, doesn’t require the axiom of choice: for every ordered set XX and function φ\varphi assigning an upper bound to each chain in XX, there’s some chain CC such that φ(C)C\varphi(C) \in C.

I also included an optional chatty section on the use of transfinite recursion to strip the isolated points from any subset of \mathbb{R}. Am I right in understanding that this is what got Cantor started on set theory in the first place?

Diagram of an ordered set, showing a chain. I know, it's not *well* ordered

Posted at 12:48 PM UTC | Permalink | Followups (7)

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.

Posted at 11:18 PM UTC | Permalink | Followups (5)

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!

Posted at 11:32 PM UTC | Permalink | Post a Comment

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 \mathbb{N}, \mathbb{Z}, \mathbb{Q} and \mathbb{R}. 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 \mathbb{N} to define sums, products and powers in \mathbb{N}, 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…

Posted at 3:08 PM UTC | Permalink | Followups (2)

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.

Posted at 5:16 AM UTC | Permalink | Followups (1)

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.

Posted at 11:15 PM UTC | Permalink | Post a Comment

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.

Posted at 2:54 PM UTC | Permalink | Followups (10)

Triangulations of the Sphere (Part 1)

Posted by John Baez

I’m writing a short column about this paper:

Let me describe one of the key ideas as simply as I can.

Posted at 12:42 AM UTC | Permalink | Followups (2)

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.

Posted at 9:53 PM UTC | Permalink | Post a Comment

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 XX by expressions like

{xX:some property of x holds}, \{ x \in X : \text{some property of }\ x\ \text{ holds} \},

and functions ff by expressions like

f(x)=some formula in x. f(x) = \text{some formula in }\ x.

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.

Quantifiers applied to a relation

Posted at 4:27 PM UTC | Permalink | Followups (2)

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:

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.

Posted at 5:00 PM UTC | Permalink | Followups (12)

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.

Posted at 11:26 AM UTC | Permalink | Followups (3)