An algebraic approach to stable domains

Day [75] showed that the category of continuous lattices and maps which preserve directed joins and arbitrary meets is the category of algebras for a monad over Set, Sp or Pos, the free functor being the set of lters of open sets. Separately, Berry [78] constructed a cartesian closed category whose morphisms preserve directed joins and connected meets, whilst Diers [79] considered similar functors independently in a study of categories of models of disjunctive theories. Girard [85] built on Berry’s work to build a new and very lean model of polymorphism. In this paper we bring these strands together, dening a monad based on lters of connected open sets and showing that its category of algebras has Berry’s (stable) morphisms and is cartesian closed. The objects have multijoins as in Diers’ work, and the slices are continuous lattices. The monad can only be dened for locally connected spaces, so via [Barr-Par

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