An algebraic approach to stable domains

Abstract

Stable domains were introduced in a restricted form by Berry to characterise sequential algorithms, and subsequently used by Girard to model polymorphism. More recently they have also been studied by Coquand, Lamarche and Winskel. Completely independently, Diers investigated them as a generalisation of the well-known Gabriel-Ulmer duality to theories such as fields which involved unique disjunction; he called the maps functors with left multiadjoint. In this paper we argue that they are naturally the algebras for connected meets and directed joins, and develop a rich algebraic theory from this. We begin with a survey of the many strands of research which are brought together in this topic. Besides the main lines of polymorphism and disjunctive logic, we find that the same domains (but with continuous maps) arise in Jung's classification of Cartesian closed categories of algebraic posets. Also, in this paper we make use of locally connected spaces, which have already been linked with indexed products and exponentials by Barr and Diaconescu. The major part of the paper is concerned with defining a monad on a category of spaces and continuous maps and showing that the category of algebras has as object posets with directed joins such that every principal lower set is a continuous lattice, and as morphisms maps which preserve connected meets and directed joins. This is precisely analogous to Day's result for continuous lattices: in this case the functor gives filters of connected open sets and can only be defined over locally connected spaces. In the final section we prove that the category is Cartesian closed. Berry and Girard do this for their categories using the trace: an important concept also exploited by Diers to give an abstract notion of spectrum with many examples in ring theory and elsewhere. In general this gives rise to a factorisation system, as will be shown in another paper. Here we use a purely equational technique, and it is very remarkable that this works because we have to show that directed joins and codirected meets commute; we are saved from this classic error in Analysis by the Berry order.