Injective spaces and function spaces
by
Martin Hotzel Escardo
Imperial College

Given an injective space D (a continuous lattice endowed with its Scott topology) and a subspace embedding j : X --> Y, Dana Scott asked whether the function [X --> D] --> [Y --> D] which takes a continuous map f : X --> D to its greatest continuous extension g : Y --> D along j is Scott continuous. In this case the extension map is a subspace embedding. We show that the extension map is Scott continuous iff D is the trivial one-point space or j is a proper map in the sense of K.H. Hofmann and Jimmie Lawson.

In order to avoid the ambiguous expression "proper subspace embedding", we refer to proper maps as finitary maps (following B. Banaschewski). We show that the finitary sober subspaces of the injective spaces are exactly the stably locally compact spaces. Moreover, the injective spaces over finitary embeddings are the algebras of the upper power space monad on the category of sober spaces, which coincide with the retracts of upper power spaces of sober spaces (and hence are sober). In the full subcategory of locally compact sober spaces, these are known to be the continuous meet-semilattices. In the full subcategory of stably locally compact spaces these are again the continuous lattices.

The above characterization of the injective spaces over finitary embeddings is an instance of a general result on injective objects in poset-enriched categories with the structure of a KZ-monad established in the paper, which we also apply to the upper and lower power locale monads.

The above results also hold for the injective spaces over dense subspace embeddings (continuous Scott domains). Moreover, we show that every sober space has a smallest finitary DENSE sober subspace (its support). The support always contains the subspace of maximal points, and in the stably locally compact case (which includes densely injective spaces) it IS the subspace of maximal points iff that subspace is compact.

Date received: June 24, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-20.