A Characterization of Distance between 1-Bounded Compact Ultrametric Spaces through a Universal Space

The category of 1-bounded compact ultrametric spaces and non-distance increasing functions (KUM’s) have been extensively used in the semantics of concurrent programming languages. In this paper a universal space U for KUM’s is introduced, such that each KUM can be isometrically embedded in it. U consists of a suitable subset of the space of functions from [0, 1) to IN, endowed with a “prefix-based” ultrametric. U allows to characterize the distance between KUM’s in terms of the Hausdorff distance between its compact subsets. As applications, it is proved how to derive the existence of limits for Cauchy towers of spaces without using the classical categorical construction and how to find solutions of recursive domain equations inside Pnco(U).