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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)