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Boundedness And Complete Distributivity
 IV, Appl. Categ. Structures
"... . We extend the concept of constructive complete distributivity so as to make it applicable to ordered sets admitting merely bounded suprema. The KZdoctrine for bounded suprema is of some independent interest and a few results about it are given. The 2category of ordered sets admitting bounded ..."
Abstract

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. We extend the concept of constructive complete distributivity so as to make it applicable to ordered sets admitting merely bounded suprema. The KZdoctrine for bounded suprema is of some independent interest and a few results about it are given. The 2category of ordered sets admitting bounded suprema over which nonempty infima distribute is shown to be biequivalent to a 2category defined in terms of idempotent relations. As a corollary we obtain a simple construction of the nonnegative reals. 1. Introduction 1.1. The main theorem of [RW1] exhibited a biequivalence between the 2category of (constructively) completely distributive lattices and suppreserving arrows, and the idempotent splitting completion of the 2category of relations  relative to any base topos. Somewhat in passing in [RW1], it was pointed out that this biequivalence provides a simple construction of the closed unit interval ([0; 1]; ), namely as the ordered set of downsets for the idempotent relat...
Doctrines Whose Structure Forms A Fully Faithful Adjoint String
 Theory Appl. Categ
, 1997
"... . We pursue the definition of a KZdoctrine in terms of a fully faithful adjoint string Dd a m a dD. We give the definition in any Graycategory. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The alge ..."
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Cited by 16 (5 self)
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. We pursue the definition of a KZdoctrine in terms of a fully faithful adjoint string Dd a m a dD. We give the definition in any Graycategory. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defined as adjunctions when we start with a KZdoctrine. 1. Introduction Free cocompletions of categories under suitable classes of colimits were the motivating examples for the definition of KZdoctrines. We introduce in this paper a notstrict version of such doctrines defined via a fully faithful adjoint string. Thus, a nonstrict KZdoctrine on a 2category K consists of a normal endo homomorphism D : K \Gamma! K, and strong transformations d : 1K \Gamma! D, and m : DD \Gamma! D in such a way that Dd a m a dD forms a fully faithful adjoint string, satisfying one equation involving the unit of...