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January 7, 2022

Optimal Transport and Enriched Categories IV: Examples of Kan-type Centres

Posted by Simon Willerton

Last time we were thinking about categories enriched over ¯ +\bar{\mathbb{R}}_+, the extended non-negative reals; such enriched categories are sometimes called generalized or Lawvere metric spaces. In the context of optimal transport with cost matrix kk, thought of as a ¯ +\bar{\mathbb{R}}_+-profunctor k:𝒮k\colon \mathcal{S}\rightsquigarrow\mathcal{R} between suppliers and receivers, we were interested in the centre of the ‘Kan-type adjunction’ between enriched functor categories, which is the following:

In this post I want to give some examples of the Kan-type centre in low dimension to try to give a sense of what they look like over ¯ +\bar{\mathbb{R}}_+. Here’s the simplest kind of example we will see.

kan_centre_32.png

Posted at 1:55 PM UTC | Permalink | Post a Comment

Intercats

Posted by John Baez

The Topos Institute has a new seminar:

The talks will be streamed and also recorded on YouTube.

It’s a new seminar series on the mathematics of interacting systems, their composition, and their behavior. Split in equal parts theory and applications, we are particularly interested in category-theoretic tools to make sense of information-processing or adaptive systems, or those that stand in a ‘bidirectional’ relationship to some environment. We aim to bring together researchers from different communities, who may already be using similar-but-different tools, in order to improve our own interaction.

Posted at 5:55 AM UTC | Permalink | Post a Comment

January 2, 2022

Adjoint School 2022

Posted by John Baez

Every year since 2018 we’ve been having annual courses on applied category theory where you can do research with experts. It’s called the Adjoint School.

You can apply to be a student at the 2022 Adjoint School now, and applications are due January 29th! Go here:

Read on for more about how this works!

Posted at 6:20 AM UTC | Permalink | Post a Comment