# Final Dialgebras: From Categories to Allegories

Roland Backhouse; Paul Hoogendijk

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 33, Issue: 4-5, page 401-426
- ISSN: 0988-3754

## Access Full Article

top## Abstract

top## How to cite

topBackhouse, Roland, and Hoogendijk, Paul. "Final Dialgebras: From Categories to Allegories." RAIRO - Theoretical Informatics and Applications 33.4-5 (2010): 401-426. <http://eudml.org/doc/221963>.

@article{Backhouse2010,

abstract = {
The study of inductive and coinductive types (like finite lists and streams,
respectively) is usually conducted within the framework of category theory, which
to all intents and purposes is a theory of sets and functions between sets.
Allegory theory, an extension of category theory due to Freyd, is
better suited to modelling relations between sets as opposed to functions between sets.
The question thus arises of how to extend the standard categorical results on the
existence of final objects in categories (for example, coalgebras and products)
to their existence in allegories. The motivation is to streamline current work on
generic programming, in which the use of a relational theory rather than a
functional theory has proved to be desirable.
In this paper, we define the notion of a relational final dialgebra and prove, for an
important class of dialgebras, that a relational final dialgebra exists in an allegory
if and only if a final dialgebra exists in the underlying category of maps. Instances
subsumed by the class we consider include coalgebras and products. An important
lemma expresses bisimulations in allegorical terms and proves this equivalent to Aczel
and Mendler's categorical definition.
},

author = {Backhouse, Roland, Hoogendijk, Paul},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Programming theory; theory of datatypes; inductive type;
co-inductive type; relation algebra; category theory; allegory theory;
generic programming; polytypic programming.; coinductive types},

language = {eng},

month = {3},

number = {4-5},

pages = {401-426},

publisher = {EDP Sciences},

title = {Final Dialgebras: From Categories to Allegories},

url = {http://eudml.org/doc/221963},

volume = {33},

year = {2010},

}

TY - JOUR

AU - Backhouse, Roland

AU - Hoogendijk, Paul

TI - Final Dialgebras: From Categories to Allegories

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 33

IS - 4-5

SP - 401

EP - 426

AB -
The study of inductive and coinductive types (like finite lists and streams,
respectively) is usually conducted within the framework of category theory, which
to all intents and purposes is a theory of sets and functions between sets.
Allegory theory, an extension of category theory due to Freyd, is
better suited to modelling relations between sets as opposed to functions between sets.
The question thus arises of how to extend the standard categorical results on the
existence of final objects in categories (for example, coalgebras and products)
to their existence in allegories. The motivation is to streamline current work on
generic programming, in which the use of a relational theory rather than a
functional theory has proved to be desirable.
In this paper, we define the notion of a relational final dialgebra and prove, for an
important class of dialgebras, that a relational final dialgebra exists in an allegory
if and only if a final dialgebra exists in the underlying category of maps. Instances
subsumed by the class we consider include coalgebras and products. An important
lemma expresses bisimulations in allegorical terms and proves this equivalent to Aczel
and Mendler's categorical definition.

LA - eng

KW - Programming theory; theory of datatypes; inductive type;
co-inductive type; relation algebra; category theory; allegory theory;
generic programming; polytypic programming.; coinductive types

UR - http://eudml.org/doc/221963

ER -

## References

top- C.J. Aarts, R.C. Backhouse, P. Hoogendijk, T.S. Voermans and J. van der Woude, A relational theory of datatypes. Available via World-Wide Web at http://www.win.tue.nl/cs/wp/papers (September 1992).
- P. Aczel, Non Well-Founded Sets, Number 14 in CSLI Lecture Notes. Center for the Study of Language and Information (Stanford, California, 1988).
- P. Aczel and N. Mendler, A final coalgebra theorem, D.H. Pitt, Ed., Category Theory and Computer Science. Springer Verlag, Lecture Notes in Comput. Sci. (1989) 357-365.
- R.C. Backhouse, P. de Bruin, P. Hoogendijk, G. Malcolm, T.S. Voermans and J. van der Woude, Polynomial relators, M. Nivat, C.S. Rattray, T. Rus and G. Scollo, Eds., in Proc. of the 2nd Conference on Algebraic Methodology and Software Technology, AMAST'91. Springer-Verlag, Workshops in Computing (1992) 303-326.
- R. Bird, O. de Moor and P. Hoogendijk, Generic functional programming with types and relations. J. Funct. Programming6 (1996) 1-28. Zbl0848.68013
- R.S. Bird and O. de Moor, Algebra of Programming. Prentice-Hall International (1996). Zbl0847.68014
- H. Doornbos, Reductivity arguments and program construction. Ph.D. Thesis, Eindhoven University of Technology, Department of Mathematics and Computing Science (1996). Zbl0852.68007
- P.J. Freyd and A. Scedrov, Categories, Allegories. North-Holland (1990).
- T. Hagino, A typed lambda calculus with categorical type constructors, D.H. Pitt, A. Poigne and D.E. Rydeheard, Eds., Category Theory and Computer Science. Springer-Verlag, Lecture Notes in Comput. Sci.283 (1988) 140-57.
- P. Hoogendijk, A Generic Theory of Datatypes. Ph.D. Thesis, Department of Mathematics and Computing Science, Eindhoven University of Technology (1997). Zbl0900.68164
- P. Hoogendijk and R. Backhouse, When do datatypes commute? E. Moggi and G. Rosolini, Eds., Category Theory and Computer Science, 7th International Conference. Springer-Verlag, Lecture Notes in Comput. Sci.1290 (1997) 242-260. Zbl0884.68086
- P. Hoogendijk and O. de Moor, What is a datatype? Technical Report 96/16, Department of Mathematics and Computing Science, Eindhoven University of Technology, 1996. J. Funct. Programming, to appear.
- B. Jacobs and J. Rutten, A tutorial on (co)algebras and (co)induction. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS62 (1997) 222-259. Zbl0880.68070
- P. Jansson and J. Jeuring, PolyP - a polytypic programming language extension. In POPL '97: The 24th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. ACM Press (1997) 470-482.
- C.B. Jay, A semantics for shape. Sci. Comput. Programming25 (1995) 251-283. Zbl0853.68119
- C.B. Jay and J.R.B. Cockett, Shapely types and shape polymorphism, D. Sannella, Ed., ESOP '94: 5th European Symposium on Programming. Springer Verlag, Lecture Notes in Comput. Sci. (1994) 302-316.
- J. Jeuring, Polytypic pattern matching. In Conference Record of FPCA '95, SIGPLAN-SIGARCH-WG2.8 Conference on Functional Programming Languages and Computer Architecture (1995) 238-248.
- J. Jeuring and P. Jansson, Polytypic programming, J. Launchbury, E. Meijer and T. Sheard, Eds., Proceedings of the Second International Summer School on Advanced Functional Programming Techniques. Springer-Verlag, Lecture Notes in Comput. Sci.1129 (1996) 68-114.
- J. Lambek, A fixpoint theorem for complete categories. Math. Z.103 (1968) 151-161. Zbl0149.26105
- J. Lambek, Subequalizers. Canad. Math. Bull.13 (1970) 337-349. Zbl0201.02302
- S. MacLane, Categories for the Working Mathematician. Springer-Verlag, New York (1971). Zbl0705.18001
- L. Meertens, Calculate polytypically! H. Kuchen and S. Doaitse Swierstra, Eds., Proceedings of the Eighth International Symposium PLILP '96 Programming Languages: Implementations, Logics and Programs. Springer Verlag, Lecture Notes in Comput. Sci.1140 (1996) 1-16.
- E. Meijer, M.M. Fokkinga and R. Paterson, Functional programming with bananas, lenses, envelopes and barbed wire. In FPCA91: Functional Programming Languages and Computer Architecture. Springer-Verlag, Lecture Notes in Comput. Sci.523 (1991) 124-144.
- S.D. Swierstra and O. de Moor, Virtual data structures, H. Partsch, B. Möller and S. Schuman, Eds., Formal Program Development. Springer-Verlag, Lecture Notes in Comput. Sci.755 (1993) 355-371.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.