Triposes as a generalization of localic geometric morphisms

Abstract In Hyland et al. (1980), Hyland, Johnstone and Pitts introduced the notion of tripos for purpose organizing construction realizability toposes in a way that generalizes localic from complete Heyting algebras. (2002), one finds generalization this eliminating an unnecessary assumption (1980). The aim paper is to characterize triposes over base topos ${\cal S}$ terms so-called constant objects functors some elementary topos. Our characterization slightly different Pitts’s PhD Thesis (Pitts, 1981) motivated by fibered view geometric morphisms as described Streicher (2020). particular, we discuss question whether Set giving rise equivalent are already triposes.