Injectivity of topological categories

Abstract

Injective objects in concrete categories frequently turn out to be objects with particularly pleasant properties. Often some form of “completeness” provides a characterization of injectivity in such a category, with injective hulls achieved through certain standard completion processes. Several results during the past decade have shown that certain specific ”topological” categories are precisely the injective objects in various natural quasicategories of concrete categories, with injective hulls obtained via certain sieve constructions. When the base category is trivial, some of these results specialize to classical results in certain categories of ordered structures; e.g., the injectives in posets characterized as complete lattices, with injective hulls the MacNeille completions, and the injectives in semilattices characterized as locales, with injective hulls the locale hulls.

This paper contains two main results. The first provides a characterization of injective objects in a setting sufficiently general as to include the above mentioned characterizations as well as many others. The second theorem gives a characterization of those objects that have injective hulls, and provides a construction of the hulls as well. Corollaries of this theorem yield numerous known injective hull constructions. The second theorem uses a much stronger hypothesis than the first. That this hypothesis is indispensible follows from a result of E. Nelson on the non-existive of injective hulls of certain σ-semilattices.