Adjunctions Between Categories Of Domains

• Authors: Michael W. Mislove;Frank J. Oles
• Venue: Fundamenta Informaticae
• Year: 1995

In this paper we show that there is no left adjoint to the inclusion functor from the full subcategory 𝒞0 of Scott domains (i.e., consistently complete ω-algebraic cpo's) to 𝒮ℱ𝒫, the category of 𝒮ℱ𝒫-objects and Scott-continuous maps. We also show there is no left adjoint to the inclusion functor from 𝒞0 to any larger category of cpo's which contains a simple five-element domain. As a corollary, there is no left adjoint to the inclusion functor from 𝒞0 to the category of L-domains. We also investigate adjunctions between categories which contain 𝒞0, such as 𝒮ℱ𝒫, and sub categories of 𝒞0. Of course, it is well-known that each of the three standard power domain constructs gives rise to a left adjoint. Since the Hoare and Smyth power domains are Scott domains, we can regard each of these two adjunctions as left adjoints to inclusion functors from appropriate subcategories of 𝒞0. But, our interest here is in adjunctions for which the target of the left adjoint is a lluf subcategory of 𝒞; such a subcategory has all Scott domains as objects, but the morphisms are more restrictive than being Scott continuous. We show that three such adjunctions exist. The first two of these are based on the Smyth power domain construction. One is a left adjoint to the inclusion functor from the category 𝒞 of consistently complete algebraic cpo's and Scott-continuous maps preserving finite, non-empty infima to the category of coherent algebraic cpo's and Scott-continuous maps. The same functor has a restriction to the subcategory of coherent algebraic cpo's whose morphisms also are Lawson continuous to the lluf subcategory of 𝒞 whose morphisms are those Scott-continuous maps which preserve all non-empty infima. The last adjunction we derive is a generalization of the Hoare power domain which satisfies the property that, if D is a nondeterministic algebra, then the image of D under the left adjoint enjoys an additional semigroup structure under which the original algebra D is among the set of idempotents. In this way, we expand the Plotkin power domain 𝒫(D) over the Scott domain D into a Scott domain.