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September 2025's Entries

A Shadow of Triality?
- Sep 13, 2025
- The octonions have nontrivial inner automorphisms of order 3. Is this related to triality?

Random Past Entries

TeXnical Issues
- September 2, 2006
- Sage advice on viewing this blog and posting comments thereon.
The Space of Physical Frameworks (Part 1)
- Aug 31, 2024
- Does statistical mechanics reduce to thermodynamics in the limit where Boltzmann’s constant goes to zero?
Higher Compactness
- Jul 24, 2013
- Compactness and local compactness are the first rungs on infinite hierarchies of conditions that can be defined using higher toposes.
Topos Institute Postdoc
- Oct 19, 2021
- The Topos Institute is trying to hire a postdoc to work on polynomial functors!
Codensity Monads
- Jan 16, 2020
- Whenever you meet a functor, ask “what is its codensity monad?”
Treq Lila
- Jun 2, 2009
- I made an album Treq Lila, in which all the melodies were created by cellular automata.

Recent Comments

A Shadow of Triality? (1)
- John Baez
  - Sep 13, 2025 12:22
  - Toby Bartels: if you read this, could you remind me how to get the spacing around absolute value signs to ...

Equivalence via Surjections (30)
- David Roberts
  - Sep 12, 2025 08:22
  - Ah, my mistake! I hadn’t read the comments on the post, I think I only saw it when it was ...
- Yemon Choi
  - Sep 9, 2025 13:07
  - it’s difficult to interpret exactly what is meant, but that ‘bijection’ isn’t surjective constructively. Indeed - this point is also ...
- Graham Manuell
  - Sep 8, 2025 08:50
  - According to the usual definitions in terms of natural numbers, computable bijections always do have computable inverses. I don’t know ...
- Tom Leinster
  - Sep 4, 2025 23:16
  - “More fitting” is a kind and charming way of putting it. Thanks!
- Stéphane Desarzens
  - Sep 4, 2025 21:11
  - On p.4 of the preprint, before equation (2), would it be more fitting to write “f:a→a′f\colon a \to a' in ...

Safeguarded AI Meeting (10)
- David Roberts
  - Sep 4, 2025 06:26
  - In all cases, the line-by-line proof details presented here were constructed by the author. It seems important to point out ...
- Jeffery Winkler
  - Sep 3, 2025 06:55
  - Here is a physics paper partially written using ChatGPT. https://arxiv.org/pdf/2508.21276 This will be more common in the future. Is this ...
- Jeffery Winkler
  - Sep 3, 2025 06:27
  - Here are some more links about the connections between physics and AI. Physics Today - Nobel Prize https://pubs.aip.org/physicstoday/article-abstract/77/12/12/3320663/Nobel-Prize-highlights-neural-networks-physics 2 Physics ...
- Jeffery Winkler
  - Aug 20, 2025 16:55
  - In 1954, Frederick Brown wrote a science fiction story titled “The Answer”. It’s so short I can reproduce it here. ...
- Matteo Capucci
  - Aug 18, 2025 07:15
  - I quote the comment I left on John’s blog about this: I’m part of SGAI and I believe this is ...

What Is an n-Theory? (118)
- Mike Shulman
  - Sep 1, 2025 18:56
  - Yes, I would say that.
- David Corfield
  - Sep 1, 2025 08:02
  - Ok, thanks. I was just thinking that I was now left unsure as to the principles of taxonomy of 2-theories ...
- Mike Shulman
  - Sep 1, 2025 00:49
  - It does look a little like the 3-theory of FOL, but the restrictions that each protype depends only on two ...
- David Corfield
  - Aug 30, 2025 08:37
  - Great, thanks! …we can also consider notions of “theory” that are models of an (n+1)(n+1)-theory somewhere else. I was wondering ...
- Mike Shulman
  - Aug 29, 2025 18:51
  - In the post, I argued that the semantic 1-theories in any doctrine form a 2-category — for instance, the 2-category ...

(BT) Diversity from (LC) Diversity (4)
- Mark Meckes
  - Sep 1, 2025 11:55
  - While thinking about an upcoming talk I will give about this work, I suddenly realized what I should have titled ...
- Tom Leinster
  - Aug 5, 2025 18:39
  - Thanks for the information on what’s needed of ψ\psi. Evidently the situation is different from the one I had in ...
- Mark Meckes
  - Aug 5, 2025 18:32
  - Regarding your first question, what we do in the paper is first show that D−1D-1 is a BT diversity (using ...
- Tom Leinster
  - Aug 5, 2025 17:22
  - That’s super nice! In the remarks after Theorem 1 of your post, and also in Theorem 1.7 of your paper, ...

How to Count n-Ary Trees (8)
- Lutz Warnke
  - Aug 20, 2025 22:04
  - There are many different proofs of the Lagrange inversion (not all use Bell polynomials), which might help to illuminate the ...
- John Baez
  - Jul 25, 2025 08:34
  - I keep meaning to expand this article a bit so it shows how we get from the general formula for ...
- John Baez
  - Jul 8, 2025 18:40
  - I blogged about Aguilar and Ardila’s work here: More secrets of the associahedra, January 25, 2018. But now I am ...
- Kenneth
  - Jul 8, 2025 18:32
  - Here is perhaps one way to interpret Lagrange inversion from a different (deeper?) perspective, namely generalized permutahedra: Marcelo Aguiar and ...
- Tom Copeland
  - Jul 8, 2025 00:06
  - Should the reciprocity for the rising and falling factorials (−nk)=(−1) k(n+k−1k) \binom{-n}{k} = (-1)^k \binom{n+k-1}{k} arise in your machinations and/or ...

2-Rig Conjectures Proved? (6)
- John Baez
  - Aug 10, 2025 12:55
  - Hey Mike: Kevin Coulembier seems to have found away around your problem—without agreeing that finding a way around it is ...
- John Baez
  - Jul 26, 2025 09:17
  - What we’re doing is sort of a categorification of that. A 2-rig contains some objects xx where the symmetry isomorphism ...
- John Baez
  - Jul 26, 2025 09:07
  - One way out, without changing any definitions, might be this: Show that Rep(M(n,k))\mathsf{Rep}(M(n,k)) is the free 2-rig on an object ...
- Jacques Distler
  - Jul 26, 2025 08:19
  - I know nothing about any of this. But in the more pedestrian world of ℤ 2\mathbb{Z}_2 graded-commutative algebras, there are ...
- John Baez
  - Jul 25, 2025 20:42
  - This is a great point. Unfortunately I have not yet carefully examined Coulembier’s paper, but now I have a great ...

Jack Morava (2)
- RodMcGuire
  - Aug 7, 2025 06:31
  - Aug 6: Johns Hopkins obituary
- David Corfield
  - Aug 4, 2025 15:01
  - Oh, that’s a pity! For a long time he’d comment here under the name ‘jackjohnson’ and always had highly interesting ...

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September 13, 2025

A Shadow of Triality?

Posted by John Baez

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It’s well known that you can construct the octonions using triality. One statement of triality is that $Spin(8)$ has nontrivial outer automorphisms of order 3. On the other hand, the octonions have nontrivial inner automorphisms of order 3. My question: can we deduce one of these facts from the other?

Continue reading “A Shadow of Triality?” …

Posted at 12:06 PM UTC | Permalink | Followups (1)