Universal homogeneous constraint structures and the hom-equivalence classes of weakly oligomorphic structures

TL;DR

This paper establishes new conditions for the existence of universal structures in classes of relational structures with constraints, and shows that the hom-equivalence class of any countable weakly oligomorphic structure has unique, model-complete, -categorical minimal and maximal elements.

Contribution

It introduces a category-theoretic framework for universal homogeneous structures and applies it to weakly oligomorphic structures, extending Fradfe9 theory.

Findings

Existence of -categorical universal structures under new conditions

Unique minimal and maximal elements in hom-equivalence classes of weakly oligomorphic structures

Automorphism groups of universal homogeneous objects analyzed

Abstract

We derive a new sufficient condition for the existence of {\omega}-categorical universal structures in classes of relational structures with constraints, augmenting results by Cherlin, Shelah, Chi, and Hubi\v{c}ka and Ne\v{s}et\v{r}il. Using this result we show that the hom-equivalence class of any countable weakly oligomorphic structure has up to isomorphism a unique model-complete smallest and greatest element, both of which are {\omega}-categorical. As the main tool we introduce the category of constraint structures, show the existence of universal homogeneous objects, and study their automorphism groups. All constructions rest on a category-theoretic version of Fra\"iss\'e's Theorem due to Droste and G\"obel. We derive sufficient conditions for a comma category to contain a universal homogeneous object. This research is motivated by the observation that all countable models of the theory of a weakly oligomorphic structure are hom-equivalent---a result akin to (part of) the Ryll-Nardzewski Theorem.