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 $k$, thought of as a $\bar{\mathbb{R}}_+$-profunctor $k\colon \mathcal{S}\rightsquigarrow\mathcal{R}$ 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.

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.