Solving reflexive domain equations in a category of complete metric spaces
This paper presents a technique by which solutions to reflexive domain equations can be found in a certain category of complete metric spaces. The objects in this category are the (non-empty) metric spaces and the arrows consist of two maps: an isometric embedding and a non-distance-increasing left inverse to it. The solution of the equation is constructed as a fixed point of a functor over this category associated with the equation. The fixed point obtained is the direct limit (colimit) of a convergent tower. This construction works if the functor is contracting, which roughly amounts to the condition that it maps every embedding to an even denser one. We also present two additional conditions, each of which is sufficient to ensure that the functor has a unique fixed point (up to isomorphism). Finally, for a large class of functors, including function space constructions, we show that these conditions are satisfied, so that they are guaranteed to have a unique fixed point. The techniques we use are so reminiscent of Banach's fixed-point theorem that we feel justified to speak of a category-theoretic version of it.
1980 Mathematical Subject classification68B10 68C01
1986 Computing Reviews CategoriesD.1.3 D.3.1 F.3.2
Key words and phrasesdomain equations complete metric spaces category theory converging towers contracting functors Banach's fixed-point theorem
Unable to display preview. Download preview PDF.
- [ABKR]P. America, J. de Bakker, J. Kok, J. Rutten, A Denotional Semantics of a Parallel Object-Oriented Language, Technical Report (CS-R8626), Centre for Mathematics and Computer Science, Amsterdam, 1986.Google Scholar
- [Du]J. Dugundji, Topology, Allen and Bacon, Rockleigh, N.J., 1966.Google Scholar
- [En]R. Engelking, General Topology, Polish Scientific Publishers, 1977.Google Scholar
- [Ha]H. Hahn, Reele Funktionen, Chelsea, New York, 1948.Google Scholar
- [Le]D. Lehmann, Categories for Mathematical Semantics, in: Proc. 17th IEEE Symposium on Foundations of Computer Science, 1976.Google Scholar
- [ML]S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, 1971.Google Scholar
- [Sc]D.S. Scott, Continuous Lattices, in: Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer-Verlag, 1972, pp. 97–136.Google Scholar