A General Domain Theory - Unifying Metric Space and Partial Order Domain Theory with Enriched Categories
by
Kim R. Wagner
University of Cambridge

We present a theory of convergence, completeness and completion which generalizes notions from metric spaces and preorders. We thus introduce a notion of sequence which generalizes Cauchy sequences in metric spaces and eventual chains in preorders. Also metric limits of Cauchy sequences and eventual least upper bounds of eventual chains in preorders are unified, as are complete metric spaces with chain complete preorders. Furthermore we introduce a general notion of ideal with which we unify metric completion with the ideal completion of preorders.

With this general theory we can do domain theory, and for instance prove Scott's inverse limit theorem, instrumental for solving recursive domain equations.

The general theory builds on that of enriched categories of Eilenberg, Kelly and Lawvere, but is presented in straightforward lattice-theoretic terms. The framework of enriched categories provides a unification of metric spaces and preorders, whereas our contribution is the unification of convergence, completeness and completion. One advantage of the general categorical framework is the scope of the unification. Metric spaces and preorders are just two simple examples of a whole family of structures covered by our theory.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-90.