IV - Morphisms and Functors

Summary

With the exception of certain developments in Chapter II, notably Sections II-2 and II-4, we largely refrained from using category-theoretic language (even when we used its tools in the context of Galois connections). Inevitably, we have to consider various types of functions between continuous lattices, and this is a natural point in our study to use the framework of category theory.

In Section IV-1 we discuss a duality based on the formalism of Galois connections between the categories DCPO_G and DCPO_D of all dcpos with upper and lower adjoints, respectively, as morphisms. We discuss in particular the categories INF and SUP, whose objects are complete lattices (in both cases) and whose morphisms are functions preserving arbitrary infs (respectively, sups). These categories are dual (IV-1.3). We saw as early as I-2.10 ff. that maps preserving arbitrary infs and directed sups play an important role in our theory. This leads us to consider the subcategory INF^↑ of INF. Its dual under the INF–SUP duality is denoted by SUP⁰; its morphisms are precisely characterized in IV-1.4(1)–(2), but as a category in itself, SUP⁰ plays a minor role. More important, however, are the full subcategories AL ⊆ CL ⊆ INF^↑ and AL^op ⊆ CL^op ⊆ SUP⁰, which consist of algebraic and continuous lattices, respectively.