On a bicomma object condition for KZ-doctrines

We study Kock-Zoberlein doctrines that satisfy a certain bicomma object con dition. Such KZ-doctrines we call admissible. Our investigation is mainly m otivated by the example of the symmetric monad on toposes. For an admissibl e 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 un der change of base, thus obtaining a characterization of the algebras as li near objects, and the classification of discrete fibrations. Known facts ab out 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 bagdom ain constructions. There is in domain theory an example of a different kind . (C) 1999 Elsevier Science B.V. All rights reserved. MSG: 18B25; 18C15; 54 B30; 18A32.