## June 1, 2020

### Categorical Probability and Statistics 2020

#### Posted by Tom Leinster

*Guest post by Tobias Fritz, Rory Lucyshyn-Wright, and Paolo Perrone*

As many of you will already know, we are organizing the workshop Categorical Probability and Statistics, which will take place online over the upcoming weekend, June 5–8.

The goal is to provide a platform for exchange of results and ideas between the various communities who pursue categorical approaches to probability and statistics. The talks will be hosted on Zoom and accompanied by discussions in a chat forum. The final schedule of the talks is available under the link above.

## May 30, 2020

### Online Magnitude Talk by Mark Meckes

#### Posted by Simon Willerton

For any magnitude fans out there, Mark Meckes is giving a Zoom talk at the Online Asymptotic Geometric Analysis Seminar next Saturday, June 6, 11:30AM (New York time) 4:30PM (Sheffield time).

- Magnitude and intrinsic volumes of convex bodies.

Abstract: Magnitude is an isometric invariant of metric spaces with origins in category theory. Although it is very difficult to exactly compute the magnitude of interesting subsets of Euclidean space, it can be shown that magnitude, or more precisely its behavior with respect to scaling, recovers many classical geometric invariants, such as volume, surface area, and Minkowski dimension. I will survey what is known about this, including results of Barcelo-Carbery, Gimperlein-Goffeng, Leinster, Willerton, and myself, and sketch the proof of an upper bound for the magnitude of a convex body in Euclidean space in terms of intrinsic volumes.

## May 16, 2020

### The Brauer 3-Group

#### Posted by John Baez

For reasons ultimately connected to physics I’ve been spending time learning about the Brauer 3-group. For any commutative ring $k$ there is a bicategory with

- algebras over $k$ as objects,
- bimodules as morphisms,
- bimodule homomorphisms as 2-morphisms.

This is a *monoidal* bicategory, since we can take the tensor product of algebras, and everything else gets along nicely with that.

Given any monoidal bicategory we can take its **core**: that is, the sub-monoidal bicategory where we only keep *invertible* objects (invertible up to equivalence), *invertible* morphisms (invertible up to 2-isomorphism), and *invertible* 2-morphisms. The core is a monoidal bicategory where everything is invertible in a suitably weakened sense so it’s called a **3-group**.

I’ll call the particular 3-group we get from a commutative ring $k$ its **Brauer 3-group**, and I’ll denote it as $\mathbf{Br}(k)$. It’s discussed on the $n$Lab: there it’s called the Picard 3-group of $k$ but denoted as $\mathbf{Br}(k)$.

Like any 3-group, we can think of $\mathbf{Br}(k)$ as a homotopy type with 3 nontrivial homotopy groups, $\pi_1, \pi_2,$ and $\pi_3$. These groups are wonderful things in themselves:

$\pi_1$ is the Brauer group of $k$.

$\pi_2$ is the Picard group of $k$.

$\pi_3$ is the group of units of $k$.

But in general, a homotopy type contains more information than its homotopy groups. So on MathOverflow I asked if anyone knew the Postnikov invariants of the Brauer 3-group — the extra glue that binds the homotopy groups together. In theory these could give extra information about our commutative ring $k$.

But Jacob Lurie said the Postnikov invariants are trivial in this case.

## May 10, 2020

### In Further Praise of Dependent Types

#### Posted by David Corfield

After an exciting foray into the rarefied atmosphere of current geometry, I want to return to something more prosaic – dependent types – the topic treated in Chapter 2 of my book. We have already had a paean on this subject a few years ago in Mike’s post – In Praise of Dependent Types, hence the title of this one.

I’ve just watched Kevin Buzzard’s talk – Is HoTT the way to do mathematics?. Kevin is a number theorist at Imperial College London who’s looking to train his undergraduates to produce computer-checked proofs of mainstream theory (e.g., theorems in algebraic geometry concerning rings and schemes) in the Lean theorem-prover.

Why Lean? Well, at (12:14) in the talk Kevin claims of mathematicians that

They use dependent types, even though they don’t know they are using dependent types.

Let’s hope they receive this news with the delight of Molière’s Mr. Jourdain:

« Par ma foi ! il y a plus de quarante ans que je dis de la prose sans que j’en susse rien, et je vous suis le plus obligé du monde de m’avoir appris cela. »

“My faith! For more than forty years I have been speaking prose while knowing nothing of it, and I am the most obliged person in the world to you for telling me so.”

## April 28, 2020

### Group Cohomology and Homotopy Fixed Points

#### Posted by John Baez

I now have a better understanding of crossed homomorphisms and why they show up so prominently in Gille and Szamuely’s *Central Simple Algebras and Galois Cohomology*. I told the tale of my enlightenment on Twitter. Basically I just read Qiaochu Yuan’s blog posts on this subject, and discovered that I’d been struggling to understand *exactly the things he had figured out and written about:*

But I didn’t say enough about what I learned, since Twitter is not so good for that. So let me do that now.

### Model Categories as a Chu Construction

#### Posted by Mike Shulman

A couple years ago I blogged about the polycategory of multivariable adjunctions and how it embeds in the 2-Chu construction $Chu(Cat,Set)$. After my talk at the ACT@UCR seminar this week, some folks were hanging out at the category theory zulip, and Reid Barton asked

The theory of model categories is self-dual. Is there some useful way to embed {model categories + Quillen adjunctions} in Chu(something) (similarly to how $Adj$ embeds in $Chu(Cat, Set)$)? Where “something” would be some flavor of “half a model category structure”, e.g., a cofibration category?

At first I thought the answer was “no”, but now I think it is “yes”. I don’t know whether the construction is good for anything, but I found it amusing, so I thought I would share.

## April 24, 2020

### Crossed Homomorphisms

#### Posted by John Baez

While reading Gille and Szamuely’s *Central Simple Algebras and Galois Cohomology* I’m finding myself frustrated by my poor understanding of $H^1$ in group cohomology.

Roughly speaking, $H^2(G,A)$ classifies group extensions of a group $G$ by an abelian group $A$ on which it acts. $H^3(G,A)$ classifies 2-group extensions of $G$ by an abelian group $A$ on which it acts. And so on —this continues on up forever. I love this story: I call it the layer-cake philosophy of cohomology. But I never figured out how $H^1$ or $H^0$ fit into this story!

If you blindly follow the pattern, $H^1(G,A)$ should classify ways of extending a group $G$ by a group $A$ on which it acts to get a 0-group. But what does that mean? Is there any way to make it make sense? There must be.

(I won’t even *try* to think about $H^0$ this way. Not today anyway.)

## April 16, 2020

### Ultracategories and 2-Monads

#### Posted by David Corfield

We began a discussion of Jacob Lurie’s Ultracategories over here, in particular whether they may be construed as algebras for some 2-monad. Perhaps this topic deserves a post to itself, rather than appearing tucked at the end of a long and fascinating discussion about condensed/pyknotic mathematics.

I have just discovered a 1995 PhD thesis by Francisco Marmolejo, advised by Robert Paré, that’s very relevant. Unfortunately the only online access is to a very poor photocopy, here. Anyway, Marmolejo characterises Makkai’s ultracategories there in a 2-monadic way. This would still leave Lurie’s somewhat differently defined ultracategories on the to-do list.

There’s then a follow-up question of characterising any such 2-monads using the codensity monad construction if possible. There’s some ongoing codensity conversation over here.

## April 15, 2020

### Online Seminar Lists

#### Posted by Tom Leinster

As you know, many online maths seminars are now running. You could fill your days with nothing but!

This short post is to note that there are now several websites where you can find very useful calendars of online seminars. Here are some of them:

This probably isn’t the first time that someone’s made a list of lists of online talks. Perhaps there are many such lists of lists. If so, you know what to do.

## April 9, 2020

### Western Hemisphere Colloquium on Geometry and Physics

#### Posted by John Baez

Another online talk series: the Western Hemisphere Colloquium on Geometry and Physics (WHCGP). This biweekly online colloquium features geometers and physicists presenting current research on a wide range of topics in the interface of the two fields. The talks are aimed at a broad audience. They will take place via Zoom on alternate Mondays at 3pm Eastern, noon Pacific, 4pm BRT. (There will be no lecture on May 25, Memorial Day in the USA.) Each session features a 60 minute talk, followed by 15 minutes for questions and discussion. You may join the meeting 15 minutes in advance. Questions and comments may be submitted to the moderator via the chat interface during the talk, or presented in person during the Q&A session. These colloquia will be recorded and will be available on the WHCGP website asap after the event.

The first two talks are:

April 13, Edward Witten (IAS), Volumes and random matrices.

April 27, Kevin Costello (PI), Topological strings, twistors, and skyrmions.

## April 6, 2020

### A Categorical View of Conditional Expectation

#### Posted by John Baez

I always like to see categories combined with probability theory and analysis. So I’m glad Prakash Panangaden did that in his talk at the ACT@UCR seminar. Afterwards we had discussions at the Category Theory Community Server, and you can see them here if you’re a member:

https://categorytheory.zulipchat.com/

You can see his slides here, or download a video here, or watch the video here.

### Structured Cospans and Petri Nets

#### Posted by John Baez

I’m giving a talk at the MIT Categories Seminar. It’ll be on Thursday April 9th, 12 noon Eastern Time.

You can see my talk live on YouTube here, with simultaneous discussion on the Category Theory Community Server. (To join this, click here; this link will expire in a while.) The talk will be recorded and remain available on YouTube.

You can already see the slides here.

### Category Theory Calendar

#### Posted by John Baez

There are now enough online events in category theory that a calendar is needed. And here it is!

It should show the times in *your* time zone, at least if you don’t prevent it from getting that information.

## March 31, 2020

### Structured Cospans and Double Categories

#### Posted by John Baez

This talk on structured cospans and double categories is the first of a two-part series; the second part is about structured cospans and Petri nets.

I gave this first talk at the ACT@UCR seminar. You can see the slides here, or download a video here, or watch the video on YouTube.

Afterwards we had discussions at the Category Theory Community Server, and you can see those here:

https://categorytheory.zulipchat.com/

(To join this, click here. This link will expire in a while.)

## March 30, 2020

### Online Worldwide Seminar on Logic and Semantics

#### Posted by John Baez

Someone should make a grand calendar, readable by everyone, of all the new math seminars that are springing into existence. Here’s another:

- Online Worldwide Seminar on Logic and Semantics, organized by Alexandra Silva, Pawel Sobocinski and Jamie Vicary.

There will be talks fortnightly at 1 pm UTC, which is currently 2 pm British Time, thanks to daylight savings time. Here are the first few:

Wednesday, April 1, — Kevin Buzzard, Imperial College London: “Is HoTT the way to do mathematics?”

Wednesday, April 15 — Joost-Pieter Katoen, Aachen University: “Termination of probabilistic programs”.

Wednesday, April 29 — Daniela Petrisan, University of Paris: “Combining probabilistic and non-deterministic choice via weak distributive laws”.

Wednesday, May 13 — Bartek Klin, Warsaw University: “Monadic monadic second order logic”.

Wednesday, May 27 — Dexter Kozen, Cornell University: “Brzozowski derivatives as distributive laws”.