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December 4, 2023
Magnitude 2023
Posted by Tom Leinster
I’m going to do something old school and live-blog a conference: Magnitude 2023, happening right now in Osaka. This is a successor to Magnitude 2019 in Edinburgh and covers all aspects of magnitude and magnitude homology, as well as hosting some talks on subjects that aren’t magnitude but feel intriguingly magnitude-adjacent.
Slides for the talks are being uploaded here.
What is magnitude? The magnitude of an enriched category is the canonical measure of its size. For instance, the magnitude of an ordinary category is the Euler characteristic of its nerve, and the magnitude of a set (as a discrete category) is its cardinality. For metric spaces, magnitude is something new, but there is a sense in which you can recover from it classical measures of size like volume, surface area and dimension.
What is magnitude homology? It’s the canonical homology theory for enriched categories. The magnitude homology of an ordinary category is the homology of its classifying space. For metric spaces, it’s something new, and has a lot to say about the existence, uniqueness and multiplicity of geodesics.
Let’s go!
December 1, 2023
Adjoint School 2024
Posted by John Baez
Are you interested in applying category-theoretic methods to problems outside of pure mathematics? Apply to the Adjoint School!
Apply here. And do it soon.
December 31, 2023. Application Due.
February - May, 2024. Learning Seminar.
June 10 - 14, 2024. In-person Research Week at the University of Oxford, UK.
Seminar on This Week’s Finds
Posted by John Baez
I wrote 300 issues of a column called This Week’s Finds, where I explained math and physics. In the fall of 2022 I gave ten talks based on these columns. I just finished giving eight more! Now I’m done.
Here you can find videos of these talks, and some lecture notes:
Topics include Young diagrams and the representation theory of classical groups, Dynkin diagrams and the classification of simple Lie groups, quaternions and octonions, the threefold way, the periodic table of -categories, the 3-strand braid group, combinatorial species, and categorifying the harmonic oscillator.
If my website dies, maybe these lectures will still survive on my YouTube playlist.
November 23, 2023
Classification of Metric Fibrations
Posted by Tom Leinster
Guest post by Yasuhiko Asao
In this blog post, I would like to introduce my recent work on metric fibrations following the preprints Magnitude and magnitude homology of filtered set enriched categories and Classification of metric fibrations.
November 13, 2023
Mathematics for Climate Change
Posted by John Baez
Some news! I’m now helping lead a new Fields Institute program on the mathematics of climate change.
October 27, 2023
Grothendieck–Galois–Brauer Theory (Part 6)
Posted by John Baez
I’ve been talking about Grothendieck’s approach to Galois theory, but I haven’t yet touched on Brauer theory. Both of these involve separable algebras, but of different kinds. For Galois theory we need commutative separable algebras, which are morally like covering spaces. For Brauer theory we’ll need separable algebras that are as noncommutative as possible, which are morally like bundles of matrix algebras. One of my ultimate goals is to unify these theories — or, just as likely, learn how someone has already done it, and explain what they did.
Both subjects are very general and conceptual. But to make sure I understand the basics, my posts so far have focused on the most classical case: separable algebras over fields. I’ve explained a few different viewpoints on them. It’s about time to move on. But before I do, I should at least classify separable algebras over fields.
October 19, 2023
The Flora Philip Fellowship
Posted by Tom Leinster
The School of Mathematics at the University of Edinburgh is pleased to invite applications for the 2023 Flora Philip Fellowship. This four-year Fellowship is specifically aimed at promising early-career postdoctoral researchers from backgrounds that are under-represented in the mathematical sciences academic community (e.g. gender, minority ethnicity, disability, disadvantaged circumstances, etc.). The Fellowship aims to provide a supportive and collegial environment for early-career researchers to develop their research and prepare themselves, with support from an academic mentor, for future independent roles in academia and beyond.
The closing date is 24 November and the job ad is here.
October 12, 2023
Grothendieck–Galois–Brauer Theory (Part 5)
Posted by John Baez
Lately I’ve been talking about ‘separable commutative algebras’, writing serious articles with actual proofs in them. Now it’s time to relax and reap the rewards! So this time I’ll come out and finally explain the geometrical meaning of separable commutative algebras.
Just so you don’t miss it, I’ll put it in boldface. And in case that’s not good enough, I’ll also say it here! Any commutative algebra gives an affine scheme called its spectrum, and is separable iff can be separated into the diagonal and the rest!
I’ll explain this better in the article.
October 1, 2023
The Free 2-Rig on One Object
Posted by John Baez
These are notes for the talk I’m giving at the Edinburgh Category Theory Seminar this Wednesday, based on work with Joe Moeller and Todd Trimble.
(No, the talk will not be recorded.)
Schur Functors
The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a universal role in representation theory, since Schur functors act on the category of representations of any group. We can understand this as an example of categorification. A ‘rig’ is a ‘ring without negatives’, and the free rig on one generator is , the rig of polynomials with natural number coefficients. Categorifying the concept of commutative rig we obtain the concept of ‘symmetric 2-rig’, and it turns out that the category of Schur functors is the free symmetric 2-rig on one generator. Thus, in a certain sense, Schur functors are the next step after polynomials.
September 28, 2023
Lectures on Applied Category Theory
Posted by John Baez
Want to learn applied category theory? You can now read my lectures here:
There are a lot, but each one is bite-sized and basically covers just one idea. They’re self-contained, but you can also read them along with Fong and Spivak’s free book to get two outlooks on the same material:
- Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.
Huge thanks go to Simon Burton for making my lectures into nice web pages! But they still need work. If you see problems, please let me know.
Here’s one problem: I need to include more of my ‘Puzzles’ in these lectures. None of the links to puzzles work. Students in the original course also wrote up answers to all of these puzzles, and to many of Fong and Spivak’s exercises. But it would take quite a bit of work to put all those into webpage form, so I can’t promise to do that. 😢
September 23, 2023
The Moduli Space of Acute Triangles
Posted by John Baez
I wrote a little article explaining the concept of ‘moduli space’ through an example. It’s due October 1st so I’d really appreciate it if you folks could take a look and see if it’s clear enough. It’s really short, and it’s written for people who know some math, but not necessarily anything about moduli spaces.
September 22, 2023
Constructing the Real Numbers as Nearly Multiplicative Sequences
Posted by Emily Riehl
I’m in Regensburg this week attending a workshop on Interactions of Proof Assistants and Mathematics. One of the lecture series is being given by John Harrison, a Senior Principal Applied Scientist in the Automated Reasoning Group at Amazon Web Services, and a lead developer of the HOL Light interactive theorem prover. He just told us about a very cool construction of the non-negative real numbers as sequences of natural numbers satisfying a property he calls “near multiplicativity”. In particular, the integers and the rational numbers aren’t needed at all! This is how the reals are constructed in HOL Light and is described in more detail in a book he wrote entitled Theorem Proving with the Real Numbers.
Edit: as the commenters note, these are also known as the Eudoxus reals and were apparently discovered by our very own Stephen Schanuel and disseminated by Ross Street. Thanks for pointing me to the history of this construction!
September 17, 2023
Counting Algebraic Structures
Posted by John Baez
The number of groups with elements goes like this, starting with :
0, 1, 1, 1, 2, 1, 2, 1, 5, …
The number of semigroups with elements goes like this:
1, 1, 5, 24, 188, 1915, 28634, 1627672, 3684030417, 105978177936292, …
Here I’m counting isomorphic guys as the same.
But how much do we know about such sequences in general? For example, is there any sort of algebraic gadget where the number of gadgets with elements goes like this:
1, 1, 2, 1, 1, 1, 1, 1, … ?
September 12, 2023
Coalgebraic Behavioural Metrics: Part 1
Posted by Emily Riehl
guest post by Keri D’Angelo, Johanna Maria Kirss and Matina Najafi and Wojtek Rozowski
Long ago, coalgebras of all kinds lived together: deterministic and nondeterministic automata, transition systems of the labelled and probabilistic varieties, finite and infinite streams, and any other arrows for an arbitrary endofunctor .
September 11, 2023
Finite Model Theory and Game Comonads: Part 2
Posted by Emily Riehl
guest post by Elena Dimitriadis, Richie Yeung, Tyler Hanks, and Zhixuan Yang
In the Part 1 of this post, we saw how logical equivalences of first-order logic (FOL) can be characterised by a combinatory game, but there are still a few unsatisfactory aspects of the formulation of EF games in Part 1:
The game was formulated in a slightly informal way, delegating the precise meaning of “turns”, “moves”, “wins” to our common sense.
There are variants of the EF game that characterise logical equivalences for other logics, but these closely related games are defined ad hoc rather than as instances of one mathematical framework.
We have confined ourselves entirely to the classical semantics of FOL in the category of sets, rather than general categorical semantics.
So you, a patron of the n-Category Café, must be thinking that category theory is perfect for addressing these problems! This is exactly what we are gonna talk about today—the framework of game comonads that was introduced by Abramsky, Dawar and Wang (2017) and Abramsky and Shah (2018).
(We will not address the third point above in this post though, but hopefully the reader will agree that what we talk about below is a useful first step towards model comparison games for general categorical logic.)