Abstract
The natural order on partial functions should be preserved by all constructions. When this Principle of Modularity is observed then increasing the domain of definition of components, or modules of such a function merely enlarges its domain, without changing any of its existing values. Conversely, ignoring the principle will typically lead to the wrong concepts for pairing (products), function objects etc.
This principle arises naturally in ordered categories, where the homsets of the category are ordered, and composition (like all constructions there) is monotone. Then the desired products, general limits and function objects are all examples of local limits and adjoints, which differ from the corresponding elementary concepts in that some of the usual equations have been replaced by inequalities.
Without further constraints the properties of the resulting limits are rather weak, e.g. local products are not unique, even up to isomorphism. Their deficiencies can be removed by lifting the concept of total function from categories of partial functions to arbitrary ordered categories.
The author is supported by SERC grants GR/E 78487 and GR/F 07866, and NSERC operating grant OGPIN 016.
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References
M. Barr, Relational algebras, in: Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 47 (Springer, 1970 ) 39–55.
J. Benabou, Introduction to bicategories,in: Lecture Notes in Mathematics 47, (Springer-Verlag, 1973) 1–77.
A. Carboni, Bicategories of partial maps, Cah. de Top. et Géom. Diff. 28 (2) (1987).
A. Carboni, S. Kasangian and R. Street, Bicategories of spans and relations, J. Pure and Appl. Alg. 33 (1984) 259–267.
A. Carboni, G.M. Kelly and R.J. Wood, A 2-categorical approach to geometric morphisms, I, Sydney Category Seminar Reports 89–19 (1989).
P.L. Curien and A. Obtulowicz, Partiality and cartesian closedness, preprint (1986).
R. diPaola and A. Heller, Dominical categories: recursion theory without elements, J. Symb. Log. 52 (1986) 594–635.
J.W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics 391 (Springer-Verlag, 1974 ).
C.A.R. Hoare and He, Jifeng, Data refinement in a categorical setting, Oxford University Computing Laboratory (1988).
C.A.R. Hoare, He Jifeng and C.E. Martin, Pre-adjunctions in order enriched categories, Oxford University Computing Laboratory (1989).
C.B. Jay, Local adjonctions, J. Pure and Appl. Alg. 53 (1988) 227–238.
C.B. Jay, Extendinging properties to categories of partial maps,LFCS Tech. Rep. 90–107.
C.B. Jay, Fixpoint and loop constructions as colimits,preprint.
G.M. Kelly and R. Street, Review of the elements of 2-categories, in: Category Seminar Sydney 1972/73, Lecture Notes in Mathematics 240 (Springer, 1974 ) 75–103.
G.M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series 64 (Cambridge University Press, 1982 ).
G. Rosolini, Continuity and effectiveness in topoi, D. Phil. thesis, University of Oxford, 1986.
G. Rosolini and E. Robinson, Categories of partial maps, Inf. and Comp. 79 (2) (1988) 95–130.
D.E. Rydeheard and J.G. Stell, Foundations of equational deduction: A categorical treatment of equational proofs and unification algorithms,in: Pitt et al, (eds), Category Theory and Computer Science,Lecture Notes in Computer Science 283 (Springer, 1987) 114–139.
M.B. Smyth and G.D. Plotkin, The category-theoretic solution of recursive domain equations, SIAM J. of Comp. 11 (1982).
R.A.G. Seely, Modelling computations: a 2-categorical framework, in: Proceedings of the Second Annual Symposium on Logic in Computer Science (1987).
B. Steffen, C.B. Jay and M. Mendier, Compositional characterization of observable program properties, Laboratory for Foundations of Computer Science, Report 89–99.
M. Wand, Fixed-point constructions in order-enriched categories, Theoretical Computer Science 8 (1979) 13–30.
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© 1991 British Computer Society
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Jay, C.B. (1991). Partial Functions, Ordered Categories, Limits and Cartesian Closure. In: Birtwistle, G. (eds) IV Higher Order Workshop, Banff 1990. Workshops in Computing. Springer, London. https://doi.org/10.1007/978-1-4471-3182-3_10
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DOI: https://doi.org/10.1007/978-1-4471-3182-3_10
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