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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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. 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...
A Basic Distributive Law
 JOURNAL OF PURE AND APPLIED ALGEBRA
, 2002
"... We pursue distributive laws between monads, particularly in the context of KZdoctrines, and show that a very basic distributive law has (constructively) completely distributive lattices for its algebras. Moreover, the resulting monad is shown to be also the double dualization monad (with respect ..."
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We pursue distributive laws between monads, particularly in the context of KZdoctrines, and show that a very basic distributive law has (constructively) completely distributive lattices for its algebras. Moreover, the resulting monad is shown to be also the double dualization monad (with respect to the subobject classifier) on ordered sets.
Information Systems for Continuous Posets
, 1993
"... The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous S ..."
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Cited by 11 (2 self)
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The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyth's "Rstructures". Various constructions on continuous posets have neat descriptions in terms of these continuous information systems; here we describe HoffmannLawson duality (which could not be done easily with Rstructures) and Vietoris power locales. 2 We also use the method to give a partial answer to a question of Johnstone's: in the context of continuous posets, Vietoris algebras are the same as localic semilattices.
Constructive complete distributivity IV
 Appl. Cat. Struct
, 1994
"... A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from downclosed 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 suppreserving arrows. There is a restrict ..."
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Cited by 7 (5 self)
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A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from downclosed 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 suppreserving arrows. There is a restriction to order ideals and "totally algebraic" lattices. Both biequivalences have left exact versions. 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 Classification Primary: 06D10 Secondary 18B35, 03G10. Keywords: completely distributive, adjunction, projective, nuclear Introduction Idempotents do not split in the category of rel...
An Adjoint Characterization of the Category of Sets
 Proc. Amer. Math. Soc
, 1994
"... If a category B with Yoneda embedding Y : B \Gamma! CAT(B op ; set) has an adjoint string, U a V a W a X a Y; then B is equivalent to set. The authors gratefully acknowledge financial support from NSERC Canada. Diagrams typeset using M. Barr's diagram macros. 1 Introduction The statement of t ..."
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If a category B with Yoneda embedding Y : B \Gamma! CAT(B op ; set) has an adjoint string, U a V a W a X a Y; then B is equivalent to set. The authors gratefully acknowledge financial support from NSERC Canada. Diagrams typeset using M. Barr's diagram macros. 1 Introduction The statement of the Abstract was implicitly conjectured in [9]. Here we establish the conjecture. We will see that it suffices to assume that B has an adjoint string V a W a X a Y with V pullback preserving. A word on foundations and our notation is necessary. We write set for the category of small sets and assume that there is a Grothendieck topos, SET, of sets which contains the set of arrows of set as an object. The 2category of category objects in SET, which we write CAT, is cartesian closed and set is an object of CAT. Thus, for C a category in CAT, CAT(C op ; set) is also an object of CAT and we abbreviate it by MC; (it was written PC in [8].) Substitution gives a 2functor M : CAT coop \Gamma...
Towards `dynamic domains': totally continuous cocomplete Qcategories, Theoret
 Comput. Sci
, 2007
"... Abstract. It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or mor ..."
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Abstract. It is common practice in both theoretical computer science and theoretical physics to describe the (static) logic of a system by means of a complete lattice. When formalizing the dynamics of such a system, the updates of that system organize themselves quite naturally in a quantale, or more generally, a quantaloid. In fact, we are lead to consider cocomplete quantaloidenriched categories as fundamental mathematical structure for a dynamic logic common to both computer science and physics. Here we explain the theory of totally continuous cocomplete categories as generalization of the wellknown theory of totally continuous suplattices. That is to say, we undertake some first steps towards a theory of “dynamic domains”.
CCD LATTICES IN PRESHEAF CATEGORIES
"... Abstract. In this paper we give a characterization of constructively completely distributive (CCD) lattices in set Cop, for C a small category with pullbacks. 1. ..."
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Abstract. In this paper we give a characterization of constructively completely distributive (CCD) lattices in set Cop, for C a small category with pullbacks. 1.
The Filter Construction Revisited
"... The lter construction, as an endofunctor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk rediscovered the construction on Heyting categories and used it, together with ..."
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The lter construction, as an endofunctor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk rediscovered the construction on Heyting categories and used it, together with E. Palmgren, to construct nonstandard models of Heyting arithmetic. In this paper we describe lter construction as a leftadjoint: applied to a leftexact category it is simply the completion of subobject semilattices under ltered (and thus all) meets. We study ltered coherent logic which is coherent logic extended to arbitrary meets and the rules that existential quantication and binary disjunction distribute over ltered meets. This logic is sound and complete for interpretations in Pitts' ltered coherent categories, and conservative over coherent logic. Restricting further to rstorder logic we show that the minimal models of Heyting arithmetic described by Moerdijk and Palmgren...
EXTENSIONS IN THE THEORY OF LAX ALGEBRAS Dedicated
"... Abstract. Recent investigations of lax algebrasin generalization of Barr's relationalalgebrasmake an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monadthere may be many such lax extensions, an ..."
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Abstract. Recent investigations of lax algebrasin generalization of Barr's relationalalgebrasmake an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monadthere may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on these lax extensions, andpresent a symptomatic situation in which distinct monads yield isomorphic categories of lax algebras.
EXTENSIONS IN THE THEORY OF LAX ALGEBRAS Dedicated to Walter Tholen on the occasion of his 60th birthday
"... Abstract. Recent investigations of lax algebras—in generalization of Barr’s relational algebras—make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monad there may be many such lax extensions, and ..."
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Abstract. Recent investigations of lax algebras—in generalization of Barr’s relational algebras—make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monad there may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on these lax extensions, and present a symptomatic situation in which distinct monads yield isomorphic categories of lax algebras. 1.