## August 31, 2019

### From Simplicial Sets to Categories

#### Posted by John Baez

There’s a well-known nerve of a category, which is a simplicial set. This defines a functor

$N \colon Cat \to sSet$

from the category of categories to the category of simplicial sets. This has a left adjoint

$F \colon sSet \to Cat$

and *this left adjoint preserves finite products*.

Do you know a published reference to a proof of the last fact? A textbook explanation would be best, but a published paper would be fine too. I don’t want you to explain the proof, because I think I understand the proof. I just need a reference.

## August 27, 2019

### Turing Categories

#### Posted by John Baez

*guest post by Georgios Bakirtzis and
Christian Williams*

We continue the Applied Category Theory Seminar with a discussion of the paper Introduction to Turing Categories by Hofstra and Cockett. Thank you to Jonathan Gallagher for all the great help in teaching our group, and to Daniel Cicala and Jules Hedges for running this seminar.

## August 16, 2019

### Graphical Regular Logic

#### Posted by John Baez

*guest post by Sophie Libkind and David Jaz Myers*

This post continues the series from the Adjoint School of Applied Category Theory 2019.

### Evil Questions About Equalizers

#### Posted by John Baez

I have a few questions about equalizers. I have my own reasons for wanting to know the answers, but I’ll admit right away that these questions are evil in the technical sense. So, investigating them requires a certain morbid curiosity… and have a feeling that some of you will be better at this than I am.

Here are the categories:

$Rex$ = [categories with finite colimits, functors preserving finite colimits]

$SMC$ = [symmetric monoidal categories, strong symmetric monoidal functors]

Both are brutally truncated stumps of very nice 2-categories!

## August 11, 2019

### Even-Dimensional Balls

#### Posted by John Baez

Some of the oddballs on the $n$-Café are interested in odd-dimensional balls, but here’s a nice thing about *even*-dimensional balls: the volume of the $2n$-dimensional ball of radius $r$ is

$\frac{(\pi r^2)^n}{n!}$

Dillon Berger pointed out that summing up over all $n$ we get

$\sum_{n=0}^\infty \frac{(\pi r^2)^n}{n!} = e^{\pi r^2}$

It looks nice. But *what does it mean?*

## August 9, 2019

### The Conway 2-Groups

#### Posted by John Baez

I recently bumped into this nice paper:

• Theo Johnson-Freyd and David Treumann, $\mathrm{H}^4(\mathrm{Co}_0,\mathbb{Z}) = \mathbb{Z}/24$.

which proves just what it says: the 4th integral cohomology of the Conway group $\mathrm{Co}_0$, in the sense of group cohomology, is $\mathbb{Z}/24$. I want to point out a few immediate consequences.

### 2020 Category Theory Conferences

#### Posted by John Baez

Here are some dates to help you plan your carbon emissions.