Abstract pre-orders

Abstract

We establish a theory of pre-orders generalized to include, among other structures, metric spaces, and strong enough to support the solution of recursive domain equations. We do this by considering pre-orders defined internally in sheaves over a complete Heyting algebra Ω. When Ω is two-valued, pre-orders internally are pre-orders externally, but when Ω is the non-negative reals with suitable Heyting algebra operations, the internal pre-orders turn into generalized ultra-metric spaces externally. In the internal logic we define the concepts of chain, convergence, completeness, compactness, etc. and it turns out that Scott's theorem about the limit/colimit coincidence of chains of complete pre-orders with mediating continuous retractions generalizes to work for any cHa Ω. The external versions of this theorem then become its well known metric variations, and corresponding theorems for as yet unexplored structures.

This article was produced while the author was on leave from Carnegie Mellon University at RISC-Linz.