Homomorphisms, Bilimits and Saturated Domains (some Very Basic Domain Theory)

The most powerful feature of categories of posets with directed sups is the ability to solve domain equations such as D ∼= D. A crucial ingredient of this technique is the fact that for certain kinds of diagrams (in particular sequences of “projection pairs”) the limit and colimit are isomorphic. This much is known to everyone in the subject: what appears not to be generally known is (i) that we only use the fact that the maps are adjoint (not that they are respectively epi and mono) and (ii) what the corresponding results for other limits (such as pullbacks) are. I propose to set out the basic definitions and results here, introducing the following terms: (i) homomorphism, for a continuous map with a left adjoint (projections being a special case), (ii) comparison, for this left adjoint (embeddings being a special case), (iii) bilimit, for the common limit and colimit of filtered diagrams of homomorphisms, (iv) bifinite, for a domain expressible as a bilimit of finite posets and (v) saturated, for a domain of which any other is a retract. I shall justify my strongly-held view that the last two should replace the existing terms “profinite” and “universal”. We begin by recalling the basic ideas of the domain-theoretic solution of equations such as D ∼= D, and showing that homomorphisms (not just projections) arise frequently. We look briefly at general limits and colimits and explain the difference between bifinite and profinite posets. Then the proof of the limit-colimit coincidence for cofiltered diagrams of homomorphisms is given. Working with cofiltered diagrams is cleaner and no more difficult than working with sequences. Although the case for sequences of projections is sufficient for solving domain equations, this general form arises naturally from “indexed retracts”. We then seek limits of other kinds of diagrams of special classes of maps, in particular pullbacks and simply-connected limits of projections. Finally we apply this to finding saturated domains.