# 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 λ*c*.λ*x.x*; ∏ *c* is the embedding of *D* into Proj(*D*).

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