Universal structures with forbidden homomorphisms
We relate the existence problem of universal objects to the properties of corresponding enriched categories (lifts or expansions). In particular, extending earlier results, we prove that for every (possibly infinite) regular set F of finite connected structures there exists a (countable) \omega-categorical universal structure U for the class Forb(F) (of all countable structures not containing any homomorphic image of a member of F). We employ a technique known as homogenization: The universal object U is the shadow (reduct) of an ultrahomogeneous structure U'. We also put the results of this paper in the context of homomorphism dualities and constraint satisfaction problems. This leads to an alternative proof of the characterization of finite dualities (given by Tardif and Ne\v{s}et\v{r}il) as well as of the characterization of infinite-finite dualities for classes of relational trees given by P. L. Erd\H{o}s, P\'{a}lv\"{o}lgyi, Tardif and Tardos. The notion of regular families of structures is motivated by the recent characterization of infinite-finite dualities for classes of relational forests (itself related to regular languages). We show how the notion of a regular family of relational trees can be extended to regular families of relational structures. This gives a partial characterization of the existence of a (countable) \omega-categorical universal object for classes Forb(F).