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.

Posted at 6:49 AM UTC | Permalink | Followups (18)

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.

Posted at 4:20 PM UTC | Permalink | Followups (15)

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.

Posted at 8:03 AM UTC | Permalink | Followups (6)

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!

Posted at 7:31 AM UTC | Permalink | Followups (3)

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?

Posted at 3:13 AM UTC | Permalink | Followups (40)

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.

Posted at 8:58 AM UTC | Permalink | Followups (47)