Fraïssé Limits in Comma Categories

Fraïssé’s theorem characterizing the existence of universal homogeneous structures is a cornerstone of model theory. A categorical version of these results was developed by Droste and Göbel. Such an abstract version of Fraïssé theory allows to construct unusual objects that are far away from the usual structures. In this paper we are going to derive sufficient conditions for a comma category to contain universal homogeneous objects. Using this criterion, we characterize homogeneous structures that possess universal homogeneous endomorphisms. The existence of such endomorphisms helps to reduce questions about the full endomorphism monoid to the self-embedding monoid of the structure. As another application we characterize the retracts of homogeneous structures that are induced by universal homogeneous retractions. This extends previous results by Bonato, Delić, Mudrinski, Dolinka, and Kubiś.