On a bicomma object condition for KZ-doctrines Original Research Article

We study Kock–Zöberlein doctrines that satisfy a certain bicomma object condition. Such KZ-doctrines we call admissible. Our investigation is mainly motivated by the example of the symmetric monad on toposes. For an admissible KZ-doctrine, we characterize its algebras in terms of cocompleteness, and we describe its Kleisi 2-category by means of its bifibrations. We obtain in terms of bifibrations a “comprehensive” factorization of 1-cells (and 2-cells). Then we investigate admissibility when the KZ-doctrine is stable under change of base, thus obtaining a characterization of the algebras as linear objects, and the classification of discrete fibrations. Known facts about the symmetric monad are revisited, such as the Waelbroeck theorems. We obtain new results for complete spreads in topos theory. Finally, we apply the theory to the similar examples of the lower power locale and the bagdomain constructions. There is in domain theory an example of a different kind.