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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.

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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)