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A 2Categorical Approach To Change Of Base And Geometric Morphisms II
, 1998
"... We introduce a notion of equipment which generalizes the earlier notion of proarrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the Vpro arising from a suitable monoidal category V. We further exhibi ..."
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Cited by 45 (7 self)
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We introduce a notion of equipment which generalizes the earlier notion of proarrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the Vpro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2category, in such a way that arbitrary functors F: L ✲ K induce equipment arrows relF: relL ✲ relK, spnF: spnL ✲ spnK, and so on, and similarly for arbitrary monoidal functors V ✲ W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locallyfullyfaithful 2functor to the 2category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.
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 ..."
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Cited by 16 (7 self)
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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...
Container Types Categorically
, 2000
"... A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a ..."
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Cited by 12 (0 self)
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A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a container type is a relator that has membership. It is shown how this definition implies various other properties that are shared by all container types. In particular, all container types have a unique strength, and all natural transformations between container types are strong. Capsule Review Progress in a scientific dicipline is readily equated with an increase in the volume of knowledge, but the true milestones are formed by the introduction of solid, precise and usable definitions. Here you will find the first generic (`polytypic') definition of the notion of `container type', a definition that is remarkably simple and suitable for formal generic proofs (as is amply illustrated in t...
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...
Constructive Complete Distributivity III
, 1992
"... A complete lattice L is constructively completely distributive, (CCD)(L), if the sup map defined on down closed subobjects has a left adjoint. We characterize preservation of this property by left exact functors between toposes using a "logical comparison transformation". The characterization is app ..."
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Cited by 3 (3 self)
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A complete lattice L is constructively completely distributive, (CCD)(L), if the sup map defined on down closed subobjects has a left adjoint. We characterize preservation of this property by left exact functors between toposes using a "logical comparison transformation". The characterization is applied to (direct images of) geometric morphisms to show that local homeomorphisms (in particular, product functors) preserve (CCD) objects, while preserving (CCD) objects implies openness. Research partially supported by grants from NSERC Canada. Diagrams typeset using Michael Barr's diagram macros. AMS Subject Classifications: 18B35, 06D10, 03G10. Introduction A complete ordered set L is constructively completely distributive, abbreviated to (CCD)(L); if there is a left adjoint to the sup map W : DL \Gamma! L; where DL is the set of downclosed subobjects of L ordered by inclusion. The condition is equivalent to (8S ` DL) i n S j S 2 S o = n fT (S) j S 2 Sg j T 2 \PiS oj which...
Patterns and Lax Lambda Laws for Relational and Imperative Programming

"... Pointfree relation calculi have been fruitful in functional programming, but in specific applications pointwise expressions can be more convenient and comprehensible than pointfree ones. To integrate pointwise with pointfree, de Moor and Gibbons [AMAST 2000] give a relational semantics for lamb ..."
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Cited by 1 (1 self)
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Pointfree relation calculi have been fruitful in functional programming, but in specific applications pointwise expressions can be more convenient and comprehensible than pointfree ones. To integrate pointwise with pointfree, de Moor and Gibbons [AMAST 2000] give a relational semantics for lambda terms with noninjective pattern matching. Alternative semantics has
DUALITY FOR CCD LATTICES
"... Abstract. The 2category of constructively completely distributive lattices is shown to be bidual to a 2category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2category of ordered sets. 1. ..."
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Abstract. The 2category of constructively completely distributive lattices is shown to be bidual to a 2category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2category of ordered sets. 1.
A note on Cauchy completeness for preorders
"... In this paper, we study the notion of Cauchycomplete preorder in a regular category, following work in [CS86], introducing the logic of a regular category. We give a different, stronger characterization than in loc.cit. for those preorders. Using this, we provide a new construction of the Cauchyco ..."
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In this paper, we study the notion of Cauchycomplete preorder in a regular category, following work in [CS86], introducing the logic of a regular category. We give a different, stronger characterization than in loc.cit. for those preorders. Using this, we provide a new construction of the Cauchycompletion in a exact category. AMS Classification: 18A40, 03G30.