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June 2, 2023

Seminar on Applied Category Theory

Posted by David Corfield

I’m hosting a small symposium next Tuesday, 6 June, on Applied Category Theory, featuring our very own John Baez. Here’s the announcement.

The language of Category Theory has been under development since the 1940s and continues to evolve to this day. It was originally created as a formal language to capture common mathematical structures and inference methods across various branches of mathematics, and later found application outside of mathematics. By introducing arrows to mediate between objects, the language is designed to represent anything that can be perceived as a process - including processes of inference and physical processes.

The first applications of Category Theory outside of mathematics and logic were to physics and to computer science. There was also an early application in biology by Robert Rosen.

But over the past decade we have seen researchers under the banner of Applied Category Theory take on a variety of novel subjects, addressing topics which include:

causality, probabilistic reasoning, statistics, learning theory, deep neural networks, dynamical systems, information theory, database theory, natural language processing, cognition, consciousness, systems biology, genomics, epidemiology, chemical reaction networks, neuroscience, complex networks, game theory, robotics, and quantum computing.

In this hybrid seminar at the Centre for Reasoning, University of Kent, we will be hearing online from two leading practitioners. All are welcome to attend.


  • In person: KS23, Keynes College, University of Kent, Canterbury

  • Online: MS Teams link


UK time (UTC +1), Tuesday 6 June

  • 15.30-15.50 David Corfield (Kent), Introduction: Applied Category Theory from a Philosophical Point of View

  • 15.50-16.50 Toby St Clere Smithe (Topos Institute, Oxford), Understanding the Bayesian Brain with Categorical Cybernetics

  • 17.00-18.00 John Baez (UC Riverside), Applied Category Theory

Posted at 7:59 AM UTC | Permalink | Post a Comment

May 30, 2023

Galois’ Fatal Duel

Posted by John Baez

On this day in 1832, Evariste Galois died in a duel. The night before, he summarized his ideas in a letter to his friend Auguste Chevalier. Hermann Weyl later wrote “This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.”

That seems exaggerated, but within mathematics it might be true. On top of that, the backstory is really dramatic! I’d never really looked into it, until today. Let me summarize a bit from Wikipedia.

Posted at 5:16 AM UTC | Permalink | Followups (9)

May 9, 2023

Symmetric Spaces and the Tenfold Way

Posted by John Baez

I’ve finally figured out the really nice connection between Clifford algebra and symmetric spaces! I gave a talk about it, and you can watch a video.

Posted at 8:55 PM UTC | Permalink | Followups (23)

May 5, 2023

Categories for Epidemiology

Posted by John Baez

Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood, Eric Redekopp and I have been creating software for modeling the spread of disease… with the help of category theory!

Posted at 10:55 AM UTC | Permalink | Post a Comment

May 3, 2023

Metric Spaces as Enriched Categories II

Posted by Simon Willerton

In the previous post I set the scene a little for enriched category theory by implying that by working ‘over’ the category of sets is a bit like working ‘over’ the integers in algebra and sometimes it is more appropriate to have a different base category just as it is sometimes more appropriate to have a different base ring. Below, we’ll see with the case of metric spaces that changing the base category can seemingly change the flavour quite a lot.

An example which I was using for illustration in the last post was that whilst, on the one hand, you can encapsulate the group actions of a group GG via the functor category [G,𝒮et][{\mathbf{\mathcal{B}}} G, \mathbf{\mathcal{S}et}] where G{\mathbf{\mathcal{B}}} G is the one object category with GG as its set of morphisms, on the other hand, you cannot in ordinary category theory encapsulate the category of representations of AA, an algebra over the complex numbers, as a category of functors into the category of vector spaces as you might hope. Indeed in ordinary category theory you can’t really see the structure of a vector space lurking in the one-object category A{\mathbf{\mathcal{B}}} A.

In this post I’ll explain what an enriched category is and how enriched category theory can, for example, allow a natural expression for a representatation category as a functor category. I’ll go on to show, following Lawvere’s insight, how metric spaces and much metric space theory can be seen to live within the realm of enriched category theory.

I’ll finish with an afterword on my experiences and thoughts on why enriched categories should be more appreciated but aren’t!

Posted at 2:16 PM UTC | Permalink | Followups (27)

April 19, 2023

Bargain-Basement Mathematics

Posted by John Baez

The fundamental theorem of Galois theory and that fundamental theorem of algebraic geometry called the Nullstellensatz are not trivial, at least not to me. But they both have cheaper versions that really are. So right now I’m pondering the difference between ‘bargain-basement mathematics’ — results that are cheap and easy — and the more glamorous, harder to understand mathematics that often gets taught in school.

Since I’m talking about bargain-basement mathematics, I’ll do it in an elementary style, at least at first — since I want beginners to follow this! I hope experts will look the other way.

Posted at 9:32 PM UTC | Permalink | Followups (14)

April 17, 2023

Metric Spaces as Enriched Categories I

Posted by Simon Willerton

Last November I gave a talk entitled “Looking at metric spaces as enriched categories ” at the African Mathematics Seminar at the invitation of Café regular Bruce Bartlett. You may remember that John gave a seminar the month before me.

The talk was aimed at general pure mathematicians, with my main assumption being that the audience knew what a group action, a representation, a metric space and a category were.

My talk was in two halves, the first half was about enriched categories in general and how metric spaces can be viewed as enriched categories. The second half was about ‘applications’ of this that I’d been involved in, that was mainly an overview of the theory of magnitude of metric spaces and a little bit on directed tight spans because of the involvement of some African mathematicians.

I decided that I would write up the first half of the talk and that is what this post and the next post will be on. The definition of enriched category will come in the next post. Hopefully it is clear from the above that these two posts are likely to be rather basic as far as the regulars at the Café are concerned!

Posted at 12:33 PM UTC | Permalink | Followups (4)

April 12, 2023

Eulerian Magnitude Homology

Posted by Tom Leinster

Guest post by Giuliamaria Menara

Magnitude homology has been discussed extensively on this blog and definitely needs no introduction.

A lot of questions about magnitude homology have been answered and a number of possible application have been explored up to this point, but magnitude homology was never exploited for the structure analysis of a graph.

Being able to use magnitude homology to look for graph substructures seems a reasonable consequence of the definition of boundary map k,\partial_{k,\ell}. Indeed, a tuple (x 0,,x k)MC k,(x_0,\dots,x_k) \in MC_{k,\ell} is such that k,(x 0,,x k)=0\partial_{k,\ell}(x_0,\dots,x_k)=0 if for every vertex x i{x 1,,x k1}x_i \in \{x_1,\dots,x_{k-1} \} it holds that len(x i1,x i^,x i+1)<len(x i1,x i,x i+1)len(x_{i-1},\hat{x_i},x_{i+1}) \lt len (x_{i-1},x_i,x_{i+1}). In other words, if every vertex of the tuple is contained in a small enough substructure, which suggests the presence of a meaningful relationship between the rank of magnitude homology groups of a graph and the subgraph counting problem.

Posted at 9:34 AM UTC | Permalink | Followups (19)

March 22, 2023

Azimuth Project News

Posted by John Baez

I blog here and also on Azimuth. Here I tend to talk about pure math and mathematical physics. There I talk about the Azimuth Project.

Let me say a bit about how that’s been going. My original plans didn’t work as expected. But I joined forces with other people who came up with something pretty cool: a rather general software framework for scientific modeling, which explicitly uses abstractions such as categories and operads. Then we applied it to epidemiology.

This is the work of many people, so it’s hard to name them all, but I’ll talk about some.

Posted at 8:03 PM UTC | Permalink | Followups (1)

March 17, 2023

Jeffrey Morton

Posted by John Baez

When he was my grad student, Jeffrey Morton worked on categorifying the theory of Feynman diagrams, and describing extended topological quantum field theories using double categories.

He got his PhD in 2007. Later he did many other things. For example, together with Jamie Vicary, he did some cool work on categorifying the Heisenberg algebra using spans of spans of groupoids. This work still needs to be made fully rigorous—someone should try!

But this is about something else.

Posted at 8:56 PM UTC | Permalink | Followups (5)

March 9, 2023

Cloning in Classical Mechanics

Posted by John Baez

Everyone likes to talk about the no-cloning theorem in quantum mechanics: you can’t build a machine where you drop an electron in the top and two electrons in the same spin state as that one pop out below. This is connected to how the category of Hilbert spaces, with its usual tensor product, is non-cartesian.

Here are two easy versions of the no-cloning theorem. First, if the dimension of a Hilbert space HH exceeds 1 there’s no linear map that duplicates states:

Δ: H HH ψ ψψ \begin{array}{cccl} \Delta \colon & H & \to & H \otimes H \\ & \psi & \mapsto & \psi \otimes \psi \end{array}

Second, there’s also no linear way to take two copies of a quantum system and find a linear process that takes the state of the first copy and writes it onto the second, while leaving the first copy unchanged:

F: HH HH ψϕ ψψ \begin{array}{cccl} F \colon & H \otimes H & \to & H \otimes H \\ & \psi \otimes \phi & \mapsto & \psi \otimes \psi \end{array}

But what about classical mechanics?

Posted at 5:50 AM UTC | Permalink | Followups (27)

March 7, 2023

This Week’s Finds (101–150)

Posted by John Baez

Here’s another present for you!

I can’t keep cranking them out at this rate, since the next batch is 438 pages long and I need a break. Tim Hosgood has kindly LaTeXed all 300 issues of This Week’s Finds, but there are lots of little formatting glitches I need to fix — mostly coming from how my formatting when I initially wrote these was a bit sloppy. Also, I’m trying to add links to published versions of all the papers I talk about. So, it takes work — about two weeks of work for this batch.

So what did I talk about in Weeks 101–150, anyway?

Posted at 7:33 PM UTC | Permalink | Followups (12)

March 6, 2023

Philosophical Perspectives on Category Theory

Posted by David Corfield

This is the title of an online talk I’m giving to the Topos Institute this Thursday (17:00 UTC), 9 March. Brush up on your Fermat primes and you can join the Zoom meeting.

It’s a good opportunity to reflect on the many years devoted to the cause of promoting the philosophical significance of category theory. As storm clouds gather over the Humanities Division here at Kent, and inducements are offered for us to leave, the brighter future I envisage may come too late for me. But I don’t doubt that the first thrill of encountering category theory around 30 years ago was the intimation of a profound way of thinking.

For my most recent views on what we should make of the rise of category theory in mathematics, see Thomas Kuhn, Modern Mathematics and the Dynamics of Reason.

But perhaps it will be successes in Applied Category Theory that will prove to be unignorable, carried out by

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. ACT2023

Posted at 4:06 PM UTC | Permalink | Followups (2)

An Invitation to Geometric Higher Categories

Posted by David Corfield

Guest post by Christoph Dorn

While the term “geometric higher category” is new, its underlying idea is not: coherences in higher structures can be derived from (stratified) manifold topology. This idea is central to the cobordism hypothesis (and to the relation of manifold singularities and dualizability structures as previously discussed on the nn-Category Café), as well as to many other parts of modern Quantum Topology. So far, however, this close relation of manifold theory and higher category theory hasn’t been fully worked out. Geometric higher category theory aims to change that, and this blog post will sketch some of the central ideas of how it does so. A slightly more comprehensive (but blog-length-exceeding) version of this introduction to geometric higher categories can be found here:

Today, I only want to focus on two basic questions about geometric higher categories: namely, what is the idea behind the connection of geometry and higher category theory? And, what are the first ingredients needed in formalizing this connection?

What is geometric about geometric higher categories?

I would like to argue that there is a useful categorization of models of higher structures into three categories. But, I will only give one good example for my argument. The absence of other examples, however, can be taken as a problem that needs to be addressed, and as one of the motivations for studying geometric higher categories! The three categories of models that I want to consider are “geometric”, “topological” and “combinatorial” models of higher structures. Really, depending on your taste, different adjectives could have been chosen for these categories: for instance, in place of “combinatorial”, maybe you find that the adjectives “categorical” or “algebraic” are more applicable for what is to follow; and in place of “geometric”, maybe saying “manifold-stratified” would have been more descriptive.

Posted at 10:32 AM UTC | Permalink | Followups (6)

March 3, 2023

Special Relativity and the Mercator Projection

Posted by John Baez

When you look at an object zipping past you at nearly the speed of light, it looks not squashed but rotated.

This phenomenon is well known: it’s called Terrell rotation. But this paper puts a new spin on it:

Posted at 8:06 PM UTC | Permalink | Followups (9)