Solving Domain Equations in a Category of Compact Metric Spaces.
Franck van Breugel · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 1994
In order to solve domain equations over 1-bounded compact metric spaces, xed points of functors on categories of 1-bounded compact metric spaces are studied.Two categories of 1-bounded compact metric spaces are considered: KMS and KMS E .In both categories, objects are isomorphic if and only if they are isometric.As a consequence, provided that the operation of a domain equation can be extended to a functor, if the functor has a xed point then this xed point is a solution of the domain equation and vice versa.It is shown that so-called locally contractive functors on KMS and contractive functors on KMS E have xed points.Furthermore, it is shown that locally contractive functors on KMS and KMS E have at most one xed point (up to isomorphism).Hence, locally contractive functors on KMS and contractive and locally contractive functors on KMS E have unique xed points.Examples are presented of extensions of various operations to functors, a simple operation which cannot be extended to a functor, and a functor not having a xed point.Most of the results in this report are based on similar -already known -results for 1-bounded complete metric spaces.