Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-04-30T11:34:26.925Z Has data issue: false hasContentIssue false
  • English
  • Français

Pre-adjunctions in order enriched categories

Published online by Cambridge University Press:  04 March 2009

C. E. Martin ,
C. A. R. Hoare  and
He Jifeng
C. E. Martin
Affiliation:
Oxford University Computing Laboratory, Oxford, UK
C. A. R. Hoare
Affiliation:
Oxford University Computing Laboratory, Oxford, UK
He Jifeng
Affiliation:
Oxford University Computing Laboratory, Oxford, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Category theory offers a unified mathematical framework for the study of specifications and programs in a variety of styles, such as procedural, functional and concurrent. One way that these different languages may be treated uniformly is by generalising the definitions of some standard categorical concepts. In this paper we reproduce in the generalised theory analogues of some standard theorems on isomorphism, and outline their applications to programming languages.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 1 , Issue 2 , July 1991 , pp. 141 - 158
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Backhouse, R.C., de Bruin, P.J..Malcolm, G., Voermans, E. and van der Woude, J.S.P. (1990) A relational theory of types. Forthcoming, Department of Computing Science Rijksuniversiteit Groningen, PO Box 800, 9700 AV Groningen, The Netherlands.Google Scholar
Barr, M. (1970) Relational Algebras. Springer-Verlg, Lecture Notes in Mathematics 137 pp. 39–55.Google Scholar
Barr, M. and Wells, C. (1990) Category Theory for Computing Science. Prentice-Hall.Google Scholar
Betti, R. and Power, A.J. (1988) On Local Adjointness of Distributive Bicategories. BolletinoU.M.I. 7. PP. 931–947.Google Scholar
Bird, R.S. (1988) Lectures on Constructive Functional Programming. Technical Monograph PRG–69, Programming Research Group, 11 Kable Road, Oxford OX1 3QD.Google Scholar
Bunge, M. (1974) Coherent extensions and relational algebras. Trans. Amer. Math. Soc. 197 355–390.CrossRefGoogle Scholar
Carboni, A. and Kasangian, S. (1983) Bicategories of spans and relations. J. Pure Appl. Algebra 33 259–267.CrossRefGoogle Scholar
Curien, P-L. (1986) Categorical combinators, sequential algorithms and funcational programming. Research Notes in Theoretical Computer Science Vol. 1, Pitman.Google Scholar
Dijkstra, E.W. and Scholten, C.S.. (1990) Predicate Calculus and Program Semantics Texts and Monographs in Computers Science, Springer-Verlag.CrossRefGoogle Scholar
Filinski, A. (1989) Declarative Continuationgs: an Investingation of Duality in Programming Language Semantics. Springer-Verlag, Lecture Notes in Computers Science 389 224–249.Google Scholar
Gray, J.W. (1974) Formal Category Theory: Adjoiness for 2-categories. Springer-Verlag Lecture Notes in Mathematics 391.CrossRefGoogle Scholar
Hayashi, S. (1985) Ajunctions of semifunctors: categorical structures in nonextensional lambda calculus. Theor. Comput. Sci. 41 95–104.CrossRefGoogle Scholar
Herrlich, and Strecher, . (1973) Category Theory. Allyn and Bacon.Google Scholar
Hoare, C. A. R. (1989) The Varieties of Programming Language. TAPSOFT Proceedings. Springer-Verlag, Lecture Notes in Computer Science 351 1–18.Google Scholar
Hoare, C. A. R. and Jifeng, He (1990) Two-categorical semantics for programming languages. To appear.Google Scholar
Hoare, C. A. R. and Jifeng, He (1990) Data refinement in a categorical setting. To appear.Google Scholar
Jay, C.B. (1988) Local adjunctions. J. Pure ad Appl. Algebra 53 227–238.CrossRefGoogle Scholar
Jay, C.B. (1990) Extending properties to categories of partial maps. Technical Report ECS-LFCS 90–107, Laboratory for Foundations of Computer Science, The King's Buildings, Edinburgh EH9 3JZ.Google Scholar
Jones, G. and Sheeran, M. (1990) Relations and refinement in circuit design. Proceedings BCS FACS workshop on Refinement, (01 1990). Springer, to be published.Google Scholar
Kelly, G.M. and Street, R. (1974) Review of the Elements of 2-categories. Springer-Verlag Lecture Notes in Mathematics 420 75–103.Google Scholar
Lambak, J. and Scott, P.J. (1985) Introduction to a Higher Order Categorical Logic. Cambridge University Press.Google Scholar
Moor, O. de (1990) Categories, relations and dynamic programming.Google Scholar
Longo, G. and Moggi, E. (1984) Cartesian Closed Categories of Enumerations for Effective Type Structures. Springer-Verger, Lecture Notes in Computer Science 173 235–247.Google Scholar
Macdonald, S. and Stone, (1989) Soft adjunctions between 2-categories. J. Pure Appl. Algebra 60 155–203.CrossRefGoogle Scholar
Mac Lane, S. (1970) Categorices For The Working Mathematician. Springer-Verlag.Google Scholar
Martini, S. (1987) An Interval Model for Second Order lambda Calculus. Springer-Verkag, Lecture Notes in Computer Science 283 219–237.Google Scholar
Moggi, E. (1985) Categories of Partial Morphisms and the λp Calculus. Springer-Verlag, Lecture Notes in Computer Science 240 242–251.Google Scholar
Morgan, C.Robinson, K. and Gardiner, P. (1989). 0n the refinement calculus. Technical monograph PRG–70, Programming Research Group, 11 Kable Road, Oxford OXI 3QD.Google Scholar
Seely, R.A.G. (1979) Weak Adjointness in Proof Theory. Springer-Verlag, Lecture Notes in Mathematics 753 697–701.Google Scholar
Seely, R.A.G. (1987) Modelling Computations: A 2-categorical Framework Logic in Computer Science 1987, Ithaca, New York. IEEE, Publications. pp. 65–71.Google Scholar
Spivey, M. (1989) The Z Reference Manual. Prentice-Hall.Google Scholar
Spivey, M. (1990) Categories for funcational programming. Unpublished lecture notes.Google Scholar
Wagner, E. (1985) Categories, Data Types and Imperative Languages. Springer-Verlag, Lecture Notes in Computer Science 240 pp. 143–164.Google Scholar
Wiweger, A. (1984) Pre-adjunctions and λ-algebraic theories. Colloq. Math 48 (2) 153–165.CrossRefGoogle Scholar