## November 19, 2019

### Chair in Pure Mathematics at Sheffield

#### Posted by Simon Willerton

Here at the University of Sheffield we are advertising a Professorship in Pure Mathematics. The application deadline is 12 January 2020.

The research in pure maths is clustered around Topology, Number Theory, and Algebra and Geometry. People with interests allied to those of the Café are strongly encouraged to apply!

Please don’t hesitate to contact Sarah Whitehouse or me if you have any questions or would like to chat informally about this position.

## November 18, 2019

### Total Maps of Turing Categories

#### Posted by John Baez

*guest post by Adam Ó Conghaile and Diego Roque*

We continue the 2019 Applied Category Theory School with a discussion of the paper Total maps of Turing categories by Cockett, Hofstra and Hrubeš. Thank you to Jonathan Gallagher for all the great help in teaching our group, to Pieter Hofstra for suggesting and coordinating the project and to Daniel Cicala and Jules Hedges for running this seminar.

## November 16, 2019

### Geodesic Spheres and Gram Matrices

#### Posted by Tom Leinster

This is a short weekend diversion, containing nothing profound.

Any sphere in $\mathbb{R}^n$ can be given the geodesic metric, where the distance between two points is defined to be the length of the arc of a great circle between them. It’s not a big surprise that the same is true in any real inner product space $X$: for instance, the unit sphere $S(X)$ centred at the origin can be given a geodesic metric in a natural way. This defines a functor

$S: \mathbf{IPS} \to \mathbf{MS}$

from inner product spaces to metric spaces, where in both cases the maps are not-necessarily-surjective isometries.

What’s more interesting is that it’s not quite as straightforward to prove as you might imagine. One way to do it leads into the topic of Gram matrices, as I’ll explain.

## November 15, 2019

### Doubles for Monoidal Categories

#### Posted by John Baez

*guest post by Fosco Loregian and Bryce Clarke*

This post belongs to the series of the Applied Category Theory Adjoint School 2019 posts. It is the result of the work of the group “Profunctor optics” led by Bartosz Milewski, and constitutes a theoretical preliminary to the real meat, i.e. the discussion of Riley’s paper Categories of Optics by Mario Roman and Emily Pillmore (hi guys! We did our best to open you the way!)

In the following post, we first introduce you to the language of co/ends. Then we deconstruct this paper by Pastro and Street:

- Craig Pastro and Ross Street, Doubles for monoidal categories,
*Theory and Applications of Categories***21**4 (2008), 61–75.

Our goal is to make its main result (almost) straightforward: *Tambara modules* can be characterized as particular profunctors, precisely those that interact well with a monoidal action on their domain. For a fixed category $X$, these endoprofunctors form the Kleisli category of a monad on $Prof(X,X)$.

## October 30, 2019

### Weak Infinity-Categories Via Terminal Coalgebras

#### Posted by Tom Leinster

A very quick post to note that Eugenia Cheng and I just published our paper Weak $\infty$-categories via terminal coalgebras in *Theory and Applications of Categories*. We, um, were not very quick about turning it around: Eugenia presented the results at Category Theory 2008 in Calais, we arXived it in 2012, and it now appears to be 2019.

Shortest possible summary: Carlos Simpson pointed out (as others may also have done) that if you consider the endofunctor $\mathcal{V} \mapsto \mathcal{V}\text{-}\mathbf{Cat}$ on the category of categories with finite products, its terminal coalgebra is $\mathbf{Str}\infty\mathbf{Cat}$, the category of *strict* $\infty$-categories. Eugenia and I show how to adapt that observation to get *weak* $\infty$-categories. Enjoy!

## October 29, 2019

### Mixing Internalization with Virtualization: a Terminological Problem

#### Posted by Mike Shulman

I used to complain sometimes that $n$-categories (including $(\infty,n)$-categories) get all the press at the expense of other higher categorical structures, but thankfully I don’t think that’s true any more (to the extent that it ever was). Double categories, “virtual” (multicategory-like) structures, and other higher-categorical notions are increasingly recognized as useful alongside the more traditional $n$-categories. However, this larger zoo of categorical structures brings with it problems of terminology: how can we consistently name all of these things?

## October 21, 2019

### Screw Theory

#### Posted by John Baez

‘Screw theory’ was invented by a guy named Ball. There should be a joke in there somewhere. But what is screw theory?

## October 19, 2019

### Jaynes on Clever Tricks

#### Posted by Tom Leinster

I’ve posted before about the eminently browsable, infuriating, provocative, inspiring, opinionated, visionary, bracing, occasionally funny, unfinished book Probability Theory: The Logic of Science that arch-Bayesian apostle of maximum entropy Edwin Jaynes was still writing when he died.

All I want to do now is to share section 8.12.4 of that book, *Clever tricks and gamesmanship*, which acts as balm for anyone who feels they’re not good at “tricks”. The rest of this post consists of that section, verbatim.

## October 14, 2019

### Diversity Workshop at UCR

#### Posted by John Baez

We’re having a workshop to promote diversity in math here at UCR:

• Riverside Mathematics Workshop for Excellence and Diversity, Friday 8 November 2019, U. C. Riverside. Organized by John Baez, Weitao Chen, Edray Goins, Ami Radunskaya, and Fred Wilhelm.

It’s happening right before the applied category theory meeting, so I hope some of you can make both… especially since Eugenia Cheng will be giving a talk on Friday!

Three talks will take place in Skye Hall—home of the math department—starting at 1 pm. After this we’ll have refreshments and an hour for students to talk to the speakers. Starting at 6 pm there will be a reception across the road at the UCR Alumni Center, with food and a panel discussion on the challenges we face in promoting diversity at U.C. Riverside.

Details follow.

## October 10, 2019

### Foundations of Math and Physics One Century After Hilbert

#### Posted by John Baez

I wrote a review of this book with chapters by Penrose, Witten, Connes, Atiyah, Smolin and others:

- John Baez, review of
*Foundations of Mathematics and Physics One Century After Hilbert: New Perspectives*, edited by Joseph Kouneiher,*Notices of the American Mathematical Society***66**no. 11 (November 2019), 1690–1692.

It gave me a chance to say a bit—just a tiny bit—about the current state of fundamental physics and the foundations of mathematics.

## October 3, 2019

### Edinburgh is Hiring

#### Posted by Tom Leinster

The advertised positions are in “algebra, geometry and topology and related fields”. Category theory is specifically mentioned, as is the importance of glue:

The Hodge Institute is a large world-class group of mathematicians whose research interests lie in Algebra, Geometry and Topology and related fields such as Category Theory and Mathematical Physics. Applicants should demonstrate an outstanding research record and contribute to the productive and strong collaborative research environment in the Hodge Institute. Preference may be given to candidates who strengthen connections between different areas of research within the Hodge Institute or more broadly between the Hodge Institute and other parts of the School.

Here “School” refers to the School of Mathematics, and the Hodge Institute is just our name for the set of people within the School who work on the subjects mentioned.

We’re advertising for Lecturers or Readers. These are both permanent positions. Lecturer is where almost everyone starts, and Reader is more senior (the rough equivalent of a junior full professor in US terminology).

## September 30, 2019

### Applied Category Theory Meeting at UCR (Part 2)

#### Posted by John Baez

Joe Moeller and I have finalized the schedule of our meeting on applied category theory:

• Applied Category Theory, special session of the Fall Western Sectional Meeting of the AMS, U. C. Riverside, Riverside, California, 9–10 November 2019.
It’s going to be really cool, with talks on everything from brakes to bicategories, from quantum physics to social networks, and more — with the power of category theory as a unifying theme! Among other things, fellow *n*-Café host Mike Shulman is going to say how to get maps between symmetric monoidal bicategories from maps between symmetric monoidal double categories.

## September 16, 2019

### Partial Evaluations 2

#### Posted by John Baez

*guest post by Carmen Constantin*

This post belongs to the series of the Applied Category Theory Adjoint School 2019 posts. It is a follow-up to Martin and Brandon’s post about partial evaluations.

Here we would like to use some results by Clementino,
Hofmann, and Janelidze
to answer the following questions: *When can we compose
partial evaluations?* and more generally *When is the
partial evaluation relation transitive?*

## September 12, 2019

### Stellenbosch is Hiring

#### Posted by John Baez

*guest post by Bruce Bartlett*

The Mathematics Division at Stellenbosch University is advertising two permanent faculty positions at the level of Senior Lecturer and Professor.

Quoting from the advertisement (Senior Lecturer position, Professor position):

The Mathematical Sciences Department is responsible for teaching and research in Mathematics, Applied Mathematics and Computer Science at Stellenbosch University. The Mathematics Division is keen to strengthen its research in Algebra, Analysis, Category Theory, Combinatorics, Logic, Number Theory, and Topology. The Faculty of Science will offer a good research establishment grant for the first two years.

## September 11, 2019

### The Riemann Hypothesis (Part 3)

#### Posted by John Baez

Now I’ll say a little about the Weil Conjectures and Grothendieck’s theory of ‘motives’. I will continue trying to avoid all the technical details, to convey some general flavor of the subject without assuming much knowledge of algebraic geometry.

I *will* start using terms like ‘variety’, but not much more. If you don’t know what that means, imagine it’s a shape described by a bunch of polynomial equations… with some points at infinity tacked on if it’s a ‘projective variety’. Also, you should know that a ‘curve’ is a 1-dimensional variety, but if we’re using the complex numbers it’ll look 2-dimensional to ordinary mortal’s eyes, like this:

This guy is an example of a ‘curve of genus 2’.

Okay, maybe now you know enough algebraic geometry for this post.