Quasi-Eheresmann-Dedecker Universes

We introduce the notion of quasi-Ehresmann-Dedecker universes in quasi-set theory in order to get a framework to develop a categorical version of quasi-set theory, so characterizing the category Qset in a similar way as the category Set is obtained from (say) ZFC plus universes. The Ehresmann-Dedecker universes generalize the usual Sonner-Grothendieck universes and are more adequate for dealing with Urelemente, which is the case of quasi-set theory. This paper is just a sketch where the main ideas are presented.