Cartesian closed categories of domains and the space proj(D)

Abstract

This paper studies algebraic dcpos and their space $\left[ {D\underrightarrow {pr}D} \right]$of Scottcontinuous projections. Let ALG_⊥ be the category of algebraic dcpos with bottom and Scott-continuous maps as arrows. If C is a full cartesian closed subcategory of ALG_⊥ such that C is closed under D → $\left[ {D\underrightarrow {pr}D} \right]$, then every object D is projection-stable, i.e., im(p) is algebraic for all p ∃ $\left[ {D\underrightarrow {pr}D} \right]$. This is equivalent to assuming that all order-dense chains in K(D) are degenerate. If C contains an isomorphic copy of the flat natural numbers, then C is not closed under finitary Scott-continuous retractions. In this case, C contains an object D, such that K(D) is not a lower set in D. This puts serious constraints on the existence of cartesian closed categories closed under D → $\left[ {D\underrightarrow {pr}D} \right]$. The cartesian closed category of dI-domains, however, is closed under D → Retr(D) and D → Proj(D), where Retr(D) is the dI-domain of all stable Scott-continuous retractions in the stable order. The dI-domain Proj(D) consists of all p ∃ Retr(D) which are below the identity in the stable order. All dI-domains are projection-stable, and Proj(D) is isomorphic to the Hoare power domain of the ideal completion of the poset of complete primes of D. It is therefore a completely distributive bialgebraic lattice and all maps f: D → E into complete lattices E preserving all existing suprema have unique extensions to Proj(D), where λc.λx.x; ∏ c is the embedding of D into Proj(D).