Realizations of topological categories (Q5956931)

The author investigates the existence of categories \({\mathcal U}\) that are (concretely) universal for (concrete) categories with property \(P\), i.e., (a) \({\mathcal U}\) has \(P\) and (b) each (concrete) category with \(P\) can be fully (completely) imbedded into \({\mathcal U}\). Main results: 1. For every category \({\mathcal X}\) that allows a faithful functor into \({\mathcal S}et\) there exists a concretely universal category for all concrete categories over \({\mathcal X}\). 2. For suitable \({\mathcal X}\) there exists a universal category for all topological categories over \({\mathcal X}\) iff there is some universal class-complete lattice. The question whether such lattice exists, remains open. However: 3. There is no universal category for all countable complete lattices (considered as categories).