Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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?

Posted at 11:31 AM UTC | Permalink | Followups (6)

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.

Posted at 8:50 AM UTC | Permalink | Post a Comment

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.

Posted at 10:25 AM UTC | Permalink | Followups (23)

September 10, 2019

The Riemann Hypothesis (Part 2)

Posted by John Baez

Now let’s dig a tiny bit deeper into the Riemann Hypothesis, and the magnificent developments in algebraic geometry it has inspired. My desire to explain this all rather simply is making the story move more slowly than planned, but I guess that’s okay.

Posted at 9:01 AM UTC | Permalink | Followups (12)

September 7, 2019

The Riemann Hypothesis (Part 1)

Posted by John Baez

I’ve been trying to understand the Riemann Hypothesis a bit better. Don’t worry, I’m not trying to prove it — that’s a dangerous quest. Indeed Ricardo Pérez-Marco has a whole list of things not to do if you want to prove the Riemann Hypothesis, such as:

Don’t expect that the problem consists in resolving a single hard difficulty. In this kind of hard problem many enemies are on your way, well hidden, and waiting for you.


Don’t go for it unless you have succeeded in other serious problems. “Serious problems” means problems that have been open and well known for years. If you think that the Riemann Hypothesis will be your first major strike, you probably deserve failure.

Taken together, his pieces of advice are sufficiently discouraging that he almost could have just said “don’t try to prove the Riemann Hypothesis”.

But trying to understand what it means, and how people have proved vaguely similar conjectures — that seems like a more reasonable hobby.

In what follows I want to keep things as simple as possible, because I’m finding, as I study this stuff, that people are generally too eager to dive into technical details before sketching out ideas in a rough way. But I will skip over a lot of standard introductory stuff on the Riemann zeta function, since that’s easy to find.

Posted at 8:00 AM UTC | Permalink | Followups (17)

September 6, 2019

Homotopy Type Theory Electronic Seminar Talks

Posted by John Baez

Learn cool math without flying around making the planet hotter! The Homotopy Type Theory Electronic Seminar Talks (HoTTEST) will be returning in Fall 2019. The speakers are:

  • October 9: Andrej Bauer
  • October 23: Anders Mörtberg
  • November 6: Andrew Swan
  • November 20: Benno van den Berg
  • December 4: Christian Sattler (TBC)
  • December 11: Richard Garner

This semester, the seminar will be meeting on alternating Wednesdays at 11:30 Eastern Time. For updates and instructions how to attend, please see

The seminar is open to everyone, but some prior familiarity with homotopy type theory will be assumed.

Posted at 10:49 AM UTC | Permalink | Followups (4)

September 3, 2019

The Narratives Category Theorists Tell Themselves

Posted by David Corfield

Years ago on this blog, I was exploring the way narrative may be used to give direction to a tradition of intellectual enquiry. This eventually led to a book chapter, Narrative and the Rationality of Mathematical Practice in B. Mazur and A. Doxiades (eds), Circles Disturbed, Princeton, 2012.

Now, someone recently reading this piece has invited to me to speak at a workshop, Narrative and mathematical argument, listed here. Reflecting on what I might discuss there, I settled on the following:

The narratives category theorists tell themselves

Category theory is an attempt to provide general tools for all of mathematics. Its history, dating back to the 1940s, is characterised by ambitious attempts to reformulate branches of mathematics and even mathematics as a whole. It has since moved on to influence theoretical computer science and mathematical physics. Resistance to this movement over the years has taken the form of accusations of engaging in abstraction for abstraction’s sake. Here we explore the role of narrative in forming the self-identity of category theorists.

Posted at 10:51 AM UTC | Permalink | Followups (28)

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:CatsSet N \colon Cat \to sSet

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

F:sSetCat 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 (14)

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)

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:

RexRex = [categories with finite colimits, functors preserving finite colimits]

SMCSMC = [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 nn-Café are interested in odd-dimensional balls, but here’s a nice thing about even-dimensional balls: the volume of the 2n2n-dimensional ball of radius rr is

(πr 2) nn! \frac{(\pi r^2)^n}{n!}

Dillon Berger pointed out that summing up over all nn we get

n=0 (πr 2) nn!=e πr 2 \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, H 4(Co 0,)=/24\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 Co 0\mathrm{Co}_0, in the sense of group cohomology, is /24\mathbb{Z}/24. I want to point out a few immediate consequences.

Posted at 8:58 AM UTC | Permalink | Followups (46)

2020 Category Theory Conferences

Posted by John Baez

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

Posted at 7:19 AM UTC | Permalink | Post a Comment

July 23, 2019

Summer Meanderings About Enriched Logic

Posted by David Corfield

Reading the recently appeared article

which treats Gabriel-Ulmer and related dualities in an enriched setting, I was wondering what sense we should make of “enriched logic”.

If, for instance, we may think of ordinary Gabriel-Ulmer duality as operating between essentially algebraic theories and their categories of models, then how to think of a finitely complete 𝒱\mathcal{V}-category as a kind of enriched essentially algebraic theory?

That got me wondering about the case where 𝒱\mathcal{V} is the reals or the real interval, i.e., something along the lines of a Lawvere metric space, which led me to some recent work on continuous logic. This logic is associated with a longstanding program on continuous model theory, but it seems that the time is ripe now for category theoretic recasting, as in:

  • Simon Cho, Categorical semantics of metric spaces and continuous logic, (arXiv:1901.09077).

In this article Cho argues that the object of truth values of continuous logic is to be seen as a “continuous subobject classifier” in the sense of topos theory.

Posted at 6:47 AM UTC | Permalink | Followups (28)