Constructive complete distributivity IV

A complete lattice L is constructively completely distributive CCD when the sup arrow from down closed subobjects of L to L has a left adjoint The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of CCD lattices and sup preserving arrows There is a restriction to order ideals and totally algebraic lattices Both biequivalences have left exact ver sions As applications we characterize projective sup lattices and recover a known characterization of projective frames Also the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of CCD lattices in a topos Research partially supported by grants from NSERC Canada Diagrams typeset using Michael Barr s diagram package AMS Subject Classi cation Primary D Secondary B G