Geometric theories and databases
Steven Vickers · Cambridge University Press eBooks · 1992
Domain theoretic understanding of databases as elements of powerdomains is modified to allow multisets of records instead of sets. This is related to geometric theories and classifying toposes, and it is shown that algebraic base domains lead to algebraic categories of models in two cases analogous to the lower (Hoare) powerdomain and Gunter's mixed powerdomain. Terminology Throughout this paper, “domain” means algebraic poset – not necessarily with bottom, nor second countable. The information system theoretic account of algebraic posets fits very neatly with powerdomain constructions. Following Vickers [90], it may be that essentially the same methods work for continuous posets; but we defer treating those until we have a better understanding of the necessary generalizations to topos theory. More concretely, a domain is a preorder (information system) (D, ⊆) of tokens , and associated with it are an algebraic poset pt D of points (ideals of D; one would normally think of pt D as the domain), and a frame ΩD of opens (upper closed subsets of D; ΩD is isomorphic to the Scott topology on pt D). “Topos” always means “Grothendieck topos”, and not “elementary topos” morphisms between toposes are understood to be geometric morphisms. S , italicized, denotes the category of sets. We shall follow, usually without comment, the notation of Vickers [89], which can be taken as our standard reference for the topological and localic notions used here.