Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T06:23:53.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Lenses, fibrations and universal translations†

Published online by Cambridge University Press:  19 September 2011

MICHAEL JOHNSON ,
ROBERT ROSEBRUGH  and
R. J. WOOD
MICHAEL JOHNSON
Affiliation:
School of Mathematics and Computing, Macquarie University, Sydney, New South Wales, Australia Email: [email protected]
ROBERT ROSEBRUGH
Affiliation:
Department of Mathematics and Computer Science, Mount Allison University, Sackville, New Brunswick, Canada Email: [email protected]
R. J. WOOD
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper extends the ‘lens’ concept for view updating in Computer Science beyond the categories of sets and ordered sets. It is first shown that a constant complement view updating strategy also corresponds to a lens for a categorical database model. A variation on the lens concept called a c-lens is introduced, and shown to correspond to the categorical notion of Grothendieck opfibration. This variant guarantees a universal solution to the view update problem for functorial update processes.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Issue 1 , February 2012 , pp. 25 - 42
Copyright
Copyright © Cambridge University Press 2011

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

Bancilhon, F. and Spyratos, N. (1981) Update semantics of relational views. ACM Transactions on Database Systems 6 557–575.CrossRefGoogle Scholar
Barr, M. and Wells, C. (1995) Category theory for computing science, second edition, Prentice-Hall.Google Scholar
Barr, M. and Wells, C. (1985) Toposes, Triples and Theories. Grundlehren Math. Wiss. 278.CrossRefGoogle Scholar
Bohannon, A., Vaughan, J. and Pierce, B. (2006) Relational Lenses: A language for updatable views. In: Proceedings of the twenty-fifth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems: PODS '06, ACM.Google Scholar
Borceux, F. (1994) Handbook of Categorical Algebra 2, Cambridge University Press.Google Scholar
Diskin, Z. and Cadish, B. (1995) Algebraic graph-based approach to management of multidatabase systems. In: Motro, A. and Tennenholtz, M. (eds.) Proceedings of The Second International Workshop on Next Generation Information Technologies and Systems (NGITS '95).Google Scholar
Foster, J., Greenwald, M., Moore, J., Pierce, B. and Schmitt, A. (2007) Combinators for bi-directional tree transformations: A linguistic approach to the view update problem. ACM Transactions on Programming Languages and Systems 29.CrossRefGoogle Scholar
Gottlob, G., Paolini, P. and Zicari, R. (1988) Properties and update semantics of consistent views. ACM Transactions on Database Systems 13 486–524.CrossRefGoogle Scholar
Hegner, S. J. (2004) An order-based theory of updates for closed database views. Annals of Mathematics and Artificial Intelligence 40 63–125.CrossRefGoogle Scholar
Hofmann, M. and Pierce, B. (1995) Positive subtyping. SIGPLAN–SIGACT Symposium on Principles of Programming Languages (POPL), ACM 186–197.Google Scholar
Janelidze, G. and Tholen, W. (1994) Facets of Descent I. Applied Categorical Structures 2 245–281.CrossRefGoogle Scholar
Johnson, M. and Rosebrugh, R. (2007) Fibrations and universal view updatability. Theoretical Computer Science 388 109–129.CrossRefGoogle Scholar
Johnson, M., Rosebrugh, R. and Wood, R. J. (2002) Entity-relationship-attribute designs and sketches. Theory and Applications of Categories 10 94–112.Google Scholar
Johnson, M., Rosebrugh, R. and Wood, R. J. (2010) Algebras and Update Strategies. Journal of Universal Computer Science 16 729–748.Google Scholar
O'Hearn, P. and Tennent, R. (1995) Parametricity and local variables. Journal of the ACM 42 658–709.CrossRefGoogle Scholar
Oles, F. J. (1982) A category-theoretic approach to the semantics of programming languages, Ph.D. Thesis, Syracuse University.Google Scholar
Oles, F. J. (1986) Type algebras, functor categories and block structure. In: Algebraic methods in semantics, Cambridge University Press 543–573.Google Scholar
Piessens, F. and Steegmans, E. (1995) Categorical data specifications. Theory and Applications of Categories 1 156–173.Google Scholar
Rosebrugh, R., Fletcher, R., Ranieri, V., Green, K., Rhinelander, J. and Wood, A. (2009) EASIK: An EA-Sketch Implementation Kit. Available from http://www.mta.ca/~rrosebru+ (accessed 20 August 2010).Google Scholar
Street, R. (1974) Fibrations and Yoneda's lemma in a 2-category. Springer-Verlag Lecture Notes in Mathematics 420 104–133.CrossRefGoogle Scholar