The Kechris-Pestov-Todor\vcevi\'c correspondence from the point of view of category theory

TL;DR

This paper offers a categorical reinterpretation of the Kechris-Pestov-Todorčević correspondence, aiming to prove a dual statement and explore the limits of categorical approaches in combinatorics, model theory, and topological dynamics.

Contribution

It provides a categorical framework for the KPT-correspondence and demonstrates how duality principles can derive dual results, deepening understanding of the interplay between these mathematical areas.

Findings

Categorical proof of a part of the KPT-correspondence

Dual statements derived via the Duality Principle

Insights into the limits of categorical methods in combinatorics

Abstract

The Kechris-Pestov-Todor\v{c}evi\'c correspondence (KPT-correspondence for short) is a surprising correspondence between model theory, combinatorics and topological dynamics. In this paper we present a categorical re-interpretation of (a part of) the KPT-correspondence with the aim of proving a dual statement. Our strategy is to take a "direct" result and then analyze the necessary infrastructure that makes the result true by providing a purely categorical proof of the categorical version of the result. We can then capitalize on the Duality Principle to obtain the dual statements almost for free. We believe that the dual version of the KPT-correspondence can not only provide the new insights into the interplay of combinatorial, model-theoretic and topological phenomena this correspondence binds together, but also explores the limits to which categorical treatment of combinatorial phenomena can take us.