Universal homogeneous constraint structures and the hom-equivalence classes of weakly oligomorphic structures
arXiv:1203.6086
Abstract
We derive a new sufficient condition for the existence of ω-categorical universal structures in classes of relational structures with constraints, augmenting results by Cherlin, Shelah, Chi, and Hubička and Nešetřil. 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 ω-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 Fraïssé's Theorem due to Droste and Göbel. 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.
25 pages