The Stone Gamut: A Coordinatization of Mathematics

We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete self-dual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a two-dimensional space we call the Stone gamut. The Stone gamut is coordinatized horizontally by coherence, ranging from −1 for sets to 1 for complete atomic Boolean algebras (CABA’s), and vertically by complexity of language. Complexity 0 contains only sets, CABA’s, and the inconsistent empty set. Complexity 1 admits noninteracting set-CABA pairs. The entire Stone duality menagerie of partial distributive lattices enters at complexity 2. Groups, rings, fields, graphs, and categories have all entered by level 16, and every category of relational structures and their homomorphisms eventually appears. The key is the identification of continuous functions and homomorphisms, which puts StonePontrjagin duality on a uniform basis by merging algebra and topology into a simple common framework. 1 Mathematics from matrices We organize much of mathematics into a single categoryChu of Chu spaces, or games as Lafont and Streicher have called them [LS91]. A Chu space is just a matrix that we shall denote =|, but unlike the matrices of linear algebra, which serve as representations of linear transformations, Chu spaces serve as representations of the objects of mathematics, and their essence resides in how they transform. This organization permits a general two-dimensional classification of mathematical objects that we call the Stone gamut, distributed horizontally by ∗This work was supported by ONR under grant number N00014-92-J-1974. 1“Spectrum,” the obvious candidate for this appliction, already has a standard meaning in Stone duality, namely the representation of the dual space of a lattice by its prime ideals. “A shape and vertically by diversity of entries. Along the horizontal axis we find Boolean algebras at the coherent or “tall” end, sets at the discrete or “flat” end, and all other objects in between, with finitedimensional vector spaces and complete semilattices in the middle as square matrices. In the vertical direction we find lattice structures near the bottom, binary relations higher up, groups higher still, and so on. Chu spaces were first described in enriched generality by M. Barr to his student P.-H. Chu, whose master’s thesis on Chu(V, k), the V -enriched category produced by what since came to be called the Chu construction, became the appendix to Barr’s monograph on *-autonomous categories [Bar79]. The latter subject generated no interest at the time but a decade later was recognized by Seely [See89] as furnishing Girard’s linear logic [Gir87] with a natural constructive semantics. Barr then proposed the Chu construction as a means of producing constructive models of linear logic [Bar91]. The subsequent history of Chu spaces has been one of successive weakenings of the enriching category V . Barr and Chu took V to be any symmetric monoidal closed category with pullbacks, de Paiva [dP89] and Brown and Gurr [BG90] restricted to order enrichment, and finally Lafont and Streicher banished enrichment altogether by taking V = Set [LS91] and calling the resulting objects games after von Neumann and Morgenstern. Chu spaces are indeed games, and moreover of the asynchronous kind, ideally suiting them as a model of concurrent behavior. However the term “Chu construction” predates it, and Barr has proposed to us in conversation the more concrete “Chu space,” which has the advantage over the general term “game” of requiring no additional disambiguating qualification. gamut of games” is in the “exaltation of larks” tradition.