June 13, 2021

Large Sets 3

Posted by Tom Leinster

Previously: Part 2

Inherent in set theory is the notion of well-ordered set. If you think about sets for long enough, well-orderings are bound to show up. In this post I’ll explain why, and I’ll summarize some of the fundamental facts about well-orderings — the least standard of which is an adjunction between sets and well-ordered sets.

There are no large sets this time, but what we do here will be needed in later instalments.

Posted at 8:29 PM UTC | Permalink | Followups (4)

June 11, 2021

Data Visualization Course

Posted by John Baez

Are you a student interested in data analysis and sustainability? Or maybe you know some students interested in these things?

Then check this out: my former student Nina Otter, who now teaches at UCLA and Leipzig, is offering a short course on how to analyze and present data using modern methods like topological data analysis — with sustainable fishing as an example!

Students who apply before June 15 have a chance to learn a lot of cool stuff and get paid for it!

In fact Nina has started a new institute, the DeMoS Institute. Here is the basic idea:

The institute carries out research on topics related to anti-democratic tendencies in our society, as well as on meta-scientific questions on how to make the scientific system more democratic. We believe that research must be done in the presence of those who bear their consequences. Therefore, we perform our research while at the same time implementing directly practices that promote inclusivity, interdisciplinarity, and in active engagement with society at large.

But here’s more about the course….

Posted at 12:34 AM UTC | Permalink | Followups (10)

June 10, 2021

Large Sets 2

Posted by Tom Leinster

Previously: Part 1. Next: Part 3

The world of large cardinals is inhabited by objects with names like inaccessible, ineffable and indescribable, evoking the vision of sets so large that they cannot be reached or attained by any means we have available. In this post, I’ll talk about the smallest sets that cannot be reached using the axiom of ETCS: limits.

Posted at 5:29 PM UTC | Permalink | Followups (61)

June 8, 2021

Large Sets 1

Posted by Tom Leinster

Next: Part 2

This is the first of a series of posts on how large cardinals look in categorical set theory.

My primary interest is not actually in large cardinals themselves. What I’m really interested in is exploring the hypothesis that everying in traditional, membership-based set theory that’s relevant to the rest of mathematics can be done smoothly in categorical set theory. I’m not sure this hypothesis is correct (and I suppose no one could ever be sure), which is why I use the words “hypothesis” and “explore”. But I know of no counterexample.

These posts won’t assume very much knowledge of anything. And I’ll try to stick to one topic per post. In this first one, all I’ll do is clear my throat.

Posted at 3:42 PM UTC | Permalink | Followups (50)

June 6, 2021

Optimal Transport and Enriched Categories I

Posted by Simon Willerton

Last year, in an ACT@UCR talk, I spoke about the Fenchel–Legendre transform from a category theoretic perspective and I showed how convex functions arise as a ‘profunctor nucleus’ in the context of categories enriched over the extended real numbers $[-\infty, \infty]$. At the end of the talk I gave four other examples of things which arise as profunctor nuclei, the final one of which was I put in at the last minute and labelled as “tentative”. John Baez took up the scent and asked me to explain why it was “tentative”, the answer was because I hadn’t thought about it for a while. I decided to write it up here at the Café, but the intervening year has had me concentrate on keeping our department running during the you-know-what, so this has been gestating a while!

Anyway in this series of posts I want to explain how aspects of optimal transport problems can be thought of in terms of enriched category theory, profunctors and related constructions. The genesis of this was a conversation following a comment of mine to Mike Shulman’s classic post “Equipments”, in particular Tobias Fritz pointing me to Cédric Villani’s book “Optimal transport: old and new”.

In this first first post I want to state the optimal transport problem (in the finite, discrete setting) and then describe the dual problem. I’ll end with a little digression on Kantorovich and a plug for the book Red Plenty by Francis Spufford.

Posted at 11:42 AM UTC | Permalink | Followups (10)

June 2, 2021

Schur Functors and Categorified Plethysm

Posted by John Baez

It’s done!

This paper has been 14 years in the making. So let me tell you a bit of its history, and then I’ll explain the paper itself.

Posted at 10:15 AM UTC | Permalink | Followups (73)

May 14, 2021

Structure vs. Observation

Posted by Emily Riehl

guest post by Stelios Tsampas and Amin Karamlou

Today we’ll be talking about the theory of universal algebra, and its less well-known counterpart of universal coalgebra. We’ll try to convince you that these two frameworks provide us with suitable tools for studying a fundamental duality that arises between structure and behaviour. Rather than jumping straight into the mathematical details we’ll start with a few motivating examples that arise in the setting of functional programming. We’ll talk more about the mathematics at play behind the scenes in the second half of this post.

Posted at 8:37 PM UTC | Permalink | Followups (24)

April 30, 2021

The Just Mathematics Collective

Posted by Tom Leinster

I recently learned of the Just Mathematics Collective, an “international collective of mathematicians” whose “goal is to shift the global mathematics community towards justice”.

It’s an ambitious initiative. We’ve had lots of discussions on this blog about specific places where the practice of mathematics meets real-world problems: surveillance of activist organizations, ties with violent armed groups, quandaries over funding, and so on. But JMC is looking at the big picture, encompassing all these issues and more. And the approach is nuanced: for example, rather than being crudely against the financial sector, it calls for a “reevaluation” of the relationship that we mathematicians have with it, also recognizing its potential for good.

Right now there’s a statement you can sign on the mathematical community’s ties with the NSA — the US National Security Agency, one of the world’s largest employers of mathematicians. You’ll probably find some familiar names on the list of signatories, including John Baez’s and mine.

Posted at 9:53 PM UTC | Permalink

April 20, 2021

Compositional Robotics

Posted by John Baez

A bunch of us are organizing a workshop on applications of category theory to robotics, as part of the IEEE International Conference on Robotics and Automation:

2021 Workshop on Compositional Robotics: Mathematics and Tools, online, 31 May 2021. Organized by Andrea Censi, Gioele Zardini, Jonathan Lorand, David Spivak, Brendan Fong, Nina Otter, Paolo Perrone, John Baez, Dylan Shell, Jason Kane, Alexandra Nilles, Andew Spielberg, and Emilio Frazzoli.

Submit your papers here by 21 May 2021!

Here’s the idea of the workshop….

April 16, 2021

Applied Category Theory 2021 — Call for Papers

Posted by John Baez

The deadline for submitting papers is coming up soon: May 12th.

Plans to run ACT 2021 as one of the first physical conferences post-lockdown are progressing well. Consider going to Cambridge! Financial support is available for students and junior researchers.

Posted at 2:45 PM UTC | Permalink | Followups (1)

April 13, 2021

Algebraic Closure

Posted by Tom Leinster

This semester I’ve been teaching an undergraduate course on Galois theory. It was all online, which meant a lot of work, but it was also a lot of fun: the students were great, and I got to know them individually better than I usually would.

For a category theorist, Galois theory is a constant provocation: very little is canonical or functorial, or at least, not in the obvious sense (for reasons closely related to the nontriviality of the Galois group). One important not-obviously-functorial construction is algebraic closure. We didn’t get to it in the course, but I spent a while absorbed in an expository note on it by Keith Conrad.

Proving that every field has an algebraic closure is not entirely trivial, but the proof in Conrad’s note seems easier and more obvious than the argument you’ll find in many algebra books. As he says, it’s a variant on a proof by Zorn, which he attributes to “B. Conrad” (presumably his brother Brian). It should be more widely known, and now I find myself asking: why would you prove it any other way?

What follows is a somewhat categorical take on the Conrad–Zorn proof.

Posted at 12:13 PM UTC | Permalink | Followups (20)

March 31, 2021

Can We Understand the Standard Model Using Octonions?

Posted by John Baez

I gave two talks in Latham Boyle and Kirill Krasnov’s Perimeter Institute workshop Octonions and the Standard Model.

The first talk was on Monday April 5th at noon Eastern Time. The second was exactly one week later, on Monday April 12th at noon Eastern Time.

Here they are…

Posted at 10:20 PM UTC | Permalink | Followups (33)

March 29, 2021

Native Type Theory (Part 3)

Posted by John Baez

guest post by Christian Williams

In Part 2 we described higher-order algebraic theories: categories with products and finite-order exponents, which present languages with (binding) operations, equations, and rewrites; from these we construct native type systems.

Now let’s use the wisdom of the Yoneda embedding!

Posted at 4:57 AM UTC | Permalink | Followups (35)

March 21, 2021

Native Type Theory (Part 2)

Posted by John Baez

guest post by Christian Williams

We’re continuing the story of Native Type Theory.

In Part 1 we introduced the internal language of a topos. Now, we describe the kinds of categories $T$ from which we will construct native type systems, via the internal language of the presheaf topos $\mathscr{P}T$.

Posted at 1:55 AM UTC | Permalink | Followups (5)

March 18, 2021

A Group Theory Problem

Posted by John Baez

Preparing a talk on octonions and the Standard Model, I’m struggling with a calculation in this paper:

and I’d like your help. The essence of the problem is nothing about octonions, it’s about Lie groups — and pretty simple Lie groups too, like $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$. So, there’s a good chance you can help me out. I’ll explain it.

Posted at 11:52 PM UTC | Permalink | Followups (27)