Universal homomorphisms, universal structures, and the polymorphism clones of homogeneous structures

Using a categorial version of Fra"iss'e’s theorem due to Droste and G"obel, we derive a criterion for a comma-category to have universal homogeneous objects. As a first application we give new existence result for universal structures and for \omega-categorical universal structures. As a second application we characterize the retracts of a large class of homogeneous structures, extending previous results by Bonato, Deli'c, Dolinka, and Kubi's. As a third application we show for a large class of homogeneous structures that their polymorphism clone is generated by polymorphisms of bounded arity, generalizing a classical result by Sierpi'nski that the clone of all functions on a given set is generated by its binary part. Further we study the cofinality and the Bergman property for clones and we give sufficient conditions on a homogeneous structure to have a polymorphism clone that has uncountable cofinality and the Bergman property.