Topogenous orders on forms

Abstract Departing from a categorical concept of topogenous orders defined relative to the bifibration of subobjects, we introduce and discuss topogenous orders on forms, i.e., faithful and amnestic functors. These topogenous orders are shown to include both closure and interior operators on forms. We define and study two special morphisms relative to a topogenous order, namely strict and final morphisms. We give a characterization of the two morphisms by the help of their cartesian and cocartesian liftings. Some examples from topology and algebra demonstrating our results are included.