Robert D. Rosebrugh

Constructive complete distributivity IV

A complete latticeL isconstructively completely distributive, (CCD), when the sup arrow from down-closed subobjects ofL toL 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 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.

Distributive laws and factorization

Abstract This article shows that the distributive laws of Beck in the bicategory of sets and matrices, wherein monads are categories, determine strict factorization systems on their composite monads. Conversely, it is shown that strict factorization systems on categories give rise to distributive laws. Moreover, these processes are shown to be mutually inverse in a precise sense. Strict factorization systems are shown to be the strict algebras for the 2-monad (−) 2 on the 2-category of categories. Further, an extension of the distributive law concept provides a correspondence with the classical factorization systems.

Fibrations and universal view updatability

Maintainability and modifiability of information system software can be enhanced by the provision of comprehensive support for views, since view support allows application programs to continue to operate unchanged when the underlying information system is modified. Supporting views depend upon a solution to the view update problem. This paper presents a new treatment of view updates for formally specified semantic data models based on the category theoretic sketch data model. The sketch data model has been the basis of a number of successful major information system consultancies. We define view updates by a universal property in models of the formal specification, and explain why this indeed gives a complete and correct treatment of view updatability, including a solution to the view update problem. However, a definition of updatability which is based on models causes some inconvenience in applications, so we prove that in a variety of circumstances updatability is guaranteed independently of the current model. This is done first with a very general criterion, and then for some specific cases relevant to applications. We include some details about the sketch data model, noting that it involves extensions of algebraic data specification techniques.

View Updatability Based on the Models of a Formal Specification

Information system software productivity can be increased by improving the maintainability and modifiability of the software produced. This latter in turn can be achieved by the provision of comprehensive support for views, since view support allows application programs to continue to operate unchanged when the underlying information system is modified. But, supporting views depends upon a solution to the view update problem, and proposed solutions to date have only had limited, rather than comprehensive, applicability. This paper presents a new treatment of view updates for formally specified information systems. The formal specification technique we use is based on category theory and has been the basis of a number of successful major information system consultancies. We define view updates by a universal property in a subcategory of models of the formal specification, and explain why this indeed gives a comprehensive treatment of view updatability, including a solution to the view update problem. However, a definition of updatability which is based on models causes some inconvenience in applications, so we prove that in a variety of circumstances updatability is guaranteed independently of the current model. The paper is predominantly theoretical, as it develops the theoretical basis of a formal methods technique, but the methods described here are currently being used in a large consultancy for a government Department of Health. Because the application area, information systems, is rarely treated by formal methods, we include some detail about the formal methods used. In fact they are extensions of the usual category theoretic specification techniques, and the solution to the view update problem can be seen as requiring the existence of an initial model for a specification.

A basic distributive law

Abstract We pursue distributive laws between monads, particularly in the context of KZ-doctrines, 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.

A database of categories

We describe a program which facilitates storage and manipulation of finitely-presented (FP) categories and finite-set valued functors. It allows storage, editing and recall of FP categories and functors. Several tools for testing properties of objects and arrows, and the computation of right and left Kan extensions are included. The program is written in ANSI C and is menu-based. Use of the program requires a basic knowledge of category theory.

Calculating Colimits Compositionally

We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages.

Database interoperability through state-based logical data independence

Computer supported cooperative work (CSCW) involving business-to-business transactions depends more and more upon database interoperability. The design of interbusiness CSCW when the businesses are already operating independent systems depends either upon effective reverse engineering (to properly discover the semantics underlying each organisations systems and through that to develop appropriate matches for interbusiness software), or upon sufficiently rich semantic models and good database management system support for logical data independence (to allow database updating through a logical view). This paper takes the second approach, presenting a rich semantic data model that the authors have been developing and have used successfully in a number of major consultancies, and a new approach to logical data independence and view up datability based on that model. We show how these approaches support database interoperability for business-to-business transactions and, for CSCW within an organisation, how they support federated databases.

Sketch Data Models, Relational Schema and Data Specifications

Abstract When different mathematical models are used for software analysis and development it is important to understand their relationships. When the models are truly mathematical, and when the aspects of reality that they seek to model are common, it may be possible to express their relationships in precise mathematical terms. This paper studies three mathematical models: The sketch data model, the relational data model, and the data specifications of Piessens and Steegmans, and determines their relationships mathematically and in detail. The constructions presented here answer reasonably long-standing theoretical questions, and offer techniques that promise to be practically useful in integrating data models.

An Adjoint Characterization of the Category of Sets

If a category B with Yoneda embedding Y: B CAT(BOP, set) has an adjoint string, U H V H W H X H Y, then B is equivalent to set.