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November 24, 2020
The Uniform Measure
Posted by Tom Leinster
Category theory has an excellent track record of formalizing intuitive statements of the form “this is the canonical such and such”. It has been especially effective in topology and algebra.
But what does it have to say about canonical measures? On many spaces, there is a choice of probability measure that seems canonical, or at least obvious: the first one that most people think of. For instance:
On a finite space, the obvious probability measure is the uniform one.
On a compact metric space whose isometry group acts transitively, the obvious probability measure is Haar measure.
On a subset of , the obvious probability measure is normalized Lebesgue measure (at least, assuming the subset has finite nonzero volume).
Emily Roff and I found a general recipe for assigning a canonical probability measure to a space, capturing all three examples above: arXiv:1908.11184. We call it the uniform measure. It’s categorically inspired rather than genuinely categorical, but I think it’s a nice story, and I’ll tell it now.
November 22, 2020
The Tenfold Way
Posted by John Baez
I now have a semiannual column in the Notices of the American Mathematical Society! I’m excited: it gives me a chance to write short explanations of cool math topics and get them read by up to 30,000 mathematicians. It’s sort of like This Week’s Finds on steroids.
Here’s the first one:
- The tenfold way, Notices Amer. Math. Soc. 67 (November 2020), 1599–1601.
November 18, 2020
Stellenbosch is Hiring
Posted by Simon Willerton
Guest post by Bruce Bartlett
The sunny campus of Stellenbosch University in South Africa is hiring!
We’re looking to make a permanent appointment of a mathematician at Lecturer or Senior Lecturer level.
- Job description and application webpage. Deadline 30 November 2020.
We’re a department with faculty having a variety of research interests, with functional analysis, model theory, algebra, mathematical physics, categorical algebra and number theory all being represented. I hope we will receive some applications from -Category Café patrons.
November 17, 2020
Magnitude Homology of Enriched Categories and Metric Spaces
Posted by Tom Leinster
Mike Shulman and I have just arXived the final, journal version of our paper Magnitude homology of enriched categories and metric spaces, to appear in Algebraic & Geometric Topology.
The first arXiv version appeared in 2017. There are now quite a few papers on magnitude homology of metric spaces that build on it (and therefore, ultimately, on Richard Hepworth and Simon Willerton’s very provocative work on magnitude homology of graphs).
However, that first version was more complicated than it really needed to be. We were lucky to get an extremely helpful referee, who prodded us into simplifying it. I think it’s now much more accessible and widely readable.
There’s been loads already on this blog about magnitude homology, so I won’t write more now. Enjoy!
November 11, 2020
The Categorical Origins of Lebesgue Integration, Revisited
Posted by Tom Leinster
I’ve just arXived a new paper: The categorical origins of Lebesgue integration (arXiv:2011.00412). Longtime Café readers may remember that I blogged about this stuff back in 2014, but I’ve only just written it up.
What’s it all about? There are two main theorems, which loosely are as follows:
Theorem A The Banach space has a simple universal property. This leads to a unique characterization of integration.
Theorem B The functor (finite measure spaces) (Banach spaces) has a simple universal property. This leads to a unique characterization of integration on finite measure spaces.
But there’s more! The mist has cleared on some important things since that last post back in 2014. I’ll give you the highlights.
November 10, 2020
Octonions and the Standard Model (Part 8)
Posted by John Baez
Last time I described the symmetry group of the exceptional Jordan algebra in terms of 10d Minkowski spacetime. We saw that in some sense it consists of four parts:
- the double cover of the Lorentz group in 10 dimensions,
- translations in left-handed spinor directions,
- translations in right-handed spinor directions, and
- ‘dilations’ (rescalings).
But our treatment was computational: the geometrical meaning of this decomposition was left obscure. As Atiyah said:
Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.
Now let’s renounce the devil’s bargain and try to understand the geometry! Jordan algebras are deeply connected to projective geometry, and the exceptional Jordan algebra is all about the octonionic projective plane.
November 9, 2020
Questions about Questions
Posted by David Corfield
The British philosopher R. G. Collingwood wrote
…whenever anybody states a thought in words, there are a great many more thoughts in his mind than there are expressed in his statement. Among these there are some which stand in a peculiar relation to the thought he has stated; they are not merely its context, they are its presuppositions. (An Essay on Metaphysics, 1940, pp. 21-22)
In my book I proposed that dependent type theory is well-suited to represent such presuppositions, largely via the device of its notion of ‘context’ (pp. 57-58, 92). Ideas for this thesis came in part from discussions at the Café, including that generated by this post, concerning Collingwood’s treatment of the presuppositions of a complex question. Something perhaps worth developing then is a dependent type-theoretic treatment of questions.
Collingwood writes in his Autobiography:
…a logic in which the answers are attended and the questions neglected is a false logic. (Autobiography, 1939, p. 31)
What is ordinarily meant when a proposition is called ‘true’, I thought, was this: (a) the proposition belongs to a question-and-answer complex which as a whole is ‘true’ in the proper sense of the word; (b) within this complex it is an answer to a certain question; (c) the question is what we ordinarily call a sensible or intelligent question, not a silly one, or in my terminology it ‘arises’; (d) the proposition is the ‘right’ answer to that question. (R.G. Collingwood, Autobiography, p. 38)
What do we have then on the logical treatment of questions? Well the Stanford Encylopedia of Philosophy is a natural starting point. And consulting SEP: Questions, we find the briefest mention of a couple of type-theoretic approaches, but nothing on dependent types:
- Ciardelli, I., F. Roelofsen and N. Theiler, 2017. Composing alternatives, Linguistics and Philosophy, 40 (1): 1-36.
- Cooper R., Ginzburg J. 2012. Negative Inquisitiveness and Alternatives-Based Negation. In: Aloni M., Kimmelman V., Roelofsen F., Sassoon G.W., Schulz K., Westera M. (eds) Logic, Language and Meaning. Lecture Notes in Computer Science, vol 7218. Springer, Berlin, Heidelberg. doi
If presuppositions are usefully treated by dependent type theory, then we would expect this logic to account well for questions. This is a task Aarne Ranta begins in his Type-Theoretic Grammar (OUP, 1994, pp. 137-142), which I’ll look to expand somewhat here. If anyone knows of a place where this section of his book is developed, then please let me know.
November 6, 2020
Octonions and the Standard Model (Part 7)
Posted by John Baez
Last time I explained the connection between the exceptional Jordan algebra and 10-dimensional Minkowski spacetime. Today I want to report on some work that Greg Egan, John Huerta and I did in November 2015. We figured out how to describe the big 78-dimensional symmetry group of the exceptional Jordan algebra in terms of 10d spacetime geometry!
This group is called , and we saw how it’s built up from:
- the double cover of the Lorentz group of 10d spacetime,
- the right-handed spinors in 10d spacetime,
- the left-handed spinors in 10d spacetime,
- scalars, .
This lets us chop up the Lie algebra of as follows:
or counting dimensions,
November 1, 2020
Octonions and the Standard Model (Part 6)
Posted by John Baez
One of the most interesting functions on the exceptional Jordan algebra is the determinant. The linear transformations that preserve this form the exceptional Lie group . And today I’ll show you how to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and spinors in 10d Minkowski spacetime.
October 29, 2020
Octonions and the Standard Model (Part 5)
Posted by John Baez
Last time I stated a couple of theorems connecting the gauge group of the Standard Model to the exceptional Jordan algebra. To prove them, it helps to become pretty comfortable with the exceptional Jordan algebra and its symmetries. And instead of trying to get the job done quickly, I’d prefer to proceed slowly and gently.
One reason is that while the exceptional Jordan algebra consists of self-adjoint matrices of octonions, we can think of the space of self-adjoint matrices of octonions as 10-dimensional Minkowski spacetime. So, to understand the exceptional Jordan algebra we can use facts about spinors and vectors in 10d spacetime! This is worth thinking about in its own right.
October 21, 2020
Epidemiological Modeling With Structured Cospans
Posted by John Baez
This is a wonderful development! Micah Halter and Evan Patterson have taken my work on structured cospans with Kenny Courser and open Petri nets with Jade Master, together with Joachim Kock’s whole-grain Petri nets, and turned them into a practical software tool!
Then they used that to build a tool for ‘compositional’ modeling of the spread of infectious disease. By ‘compositional’, I mean that they make it easy to build more complex models by sticking together smaller, simpler models.
Even better, they’ve illustrated the use of this tool by rebuilding part of the model that the UK has been using to make policy decisions about COVID19.
All this software was written in the programming language Julia.
October 20, 2020
No New Normed Division Algebra Found!
Posted by John Baez
Good news! The paper mentioned in my last article here, Eight-dimensional octonion-like but associative normed division algebra, has been retracted:
- Scott Chapman, Statement of retraction: Eight-dimensional octonion-like but associative normed division algebra, Communications in Algebra, 19 October 2020.
September 23, 2020
New Normed Division Algebra Found!
Posted by John Baez
Hurwitz’s theorem says that there are only 4 normed division algebras over the real numbers, up to isomorphism: the real numbers, the complex numbers, the quaternions, and the octonions. The proof was published in 1923. It’s a famous result, and several other proofs are known. I’ve spent a lot of time studying them.
Thus you can imagine my surprise today when I learned Hurwitz’s theorem was false!
- Joy Christian, Eight-dimensional octonion-like but associative normed division algebra, Communications in Algebra (2020), 1-10.
Abstract. We present an eight-dimensional even sub-algebra of the -dimensional associative Clifford algebra and show that its eight-dimensional elements denoted as and respect the norm relation , thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.
Even more wonderful is that the author has discovered that the unit vectors in his normed division algebra form a 7-sphere that is not homeomorphic to the standard 7-sphere. Exotic 7-spheres are a dime a dozen, but those merely fail to be diffeomorphic to the standard 7-sphere.
September 17, 2020
Special Numbers in Category Theory
Posted by John Baez
There are a few theorems in abstract category theory in which specific numbers play an important role. For example:
Theorem. Let be the free symmetric monoidal category on an object . Regard as a mere category. Then there exists an equivalence such that:
- is not naturally isomorphic to the identity,
- acts as the identity on all objects,
- acts as the identity on all endomorphisms except when .
This theorem would become false if we replaced by any other number.
September 16, 2020
Open Systems: A Double Categorical Perspective (Part 2)
Posted by John Baez
Back to Kenny Courser’s thesis:
- Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020.
One thing Kenny does here is explain the flaws in a well-known framework for studying open systems: decorated cospans. Decorated cospans were developed by my student Brendan Fong. Since I was Brendan’s advisor at the time, a hefty helping of blame for not noticing the problems belongs to me! But luckily, Kenny doesn’t just point out the problems: he shows how to fix them. As a result, everything we’ve done with decorated cospans can be saved.