A Functorial Approach to Dedekind Completions and the Representation of Vector Lattices and `-algebras by Normal Functions

Unlike the uniform completion, the Dedekind completion of a vector lattice is not functorial. In order to repair the lack of functoriality of Dedekind completions, we enrich the signature of vector lattices with a proximity relation, thus arriving at the category pdv of proximity Dedekind vector lattices. We prove that the Dedekind completion induces a functor from the category bav of bounded archimedean vector lattices to pdv , which in fact is an equivalence. We utilize the results of Dilworth [14] to show that every proximity Dedekind vector lattice D is represented as the normal real-valued functions on the compact Hausdorff space associated with D. This yields a contravariant adjunction between pdv and the category KHaus of compact Hausdorff spaces, which restricts to a dual equivalence between KHaus and the proper subcategory of pdv consisting of those proximity Dedekind vector lattices in which the proximity is uniformly closed. We show how to derive the classic Yosida Representation [40], Kakutani-Krein Duality [24, 26], Stone-Gelfand-Naimark Duality [35, 16], and StoneNakano Theorem [35, 32] from our approach.