Structured lattices and topological categories of L-sets

Goguen's category of V-sets whose objects are functions with values in a pre-ordered set and morphisms are suitable maps is shown to be a topological construct if and only if the pre-ordered set is a complete lattice; in particular the category Set(L) of L-sets, also considered by Goguen, is topological over Set. The special case when the considered pre-ordered set is L with the “mapping to” relation arising from a structure Φ=(ϕa)a∈L, where every ϕa:L→[⊥,a] preserves arbitrary infs, already considered by the authors on any complete lattice L, is investigated in detail.