An example of a Fra\"iss\'e class without a Kat\v{e}tov functor

TL;DR

This paper constructs a specific Fra"iss"e class that does not admit a Kat"etov functor, disproving a previous conjecture, while also demonstrating that its automorphism group remains universal.

Contribution

It provides a counterexample to the existence of Kat"etov functors for certain Fra"iss"e classes and explores the properties of the automorphism group in this context.

Findings

Existence of a Fra"iss"e class without a Kat"etov functor

Automorphism group of the Fra"iss"e limit is universal

Disproof of a conjecture from prior work

Abstract

We disprove a conjecture from [W. Kubi\'s, D. Ma\v{s}ulovi\'c, Kat\v{e}tov functors, preprint, http://arxiv.org/abs/1412.1850] by showing the existence of a Fra\"iss\'e class $\mathcal{C}$ which does not admit a Kat\v{e}tov functor. On the other hand, we show that the automorphism group of the Fra\"iss\'e limit of $\mathcal{C}$ is universal, as it happens in the presence of a Kat\v{e}tov functor.