## The Stone gamut: A coordinatization of mathematics (1995)

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Venue: | In Logic in Computer Science |

Citations: | 30 - 13 self |

### BibTeX

@INPROCEEDINGS{Pratt95thestone,

author = {Vaughan R. Pratt},

title = {The Stone gamut: A coordinatization of mathematics},

booktitle = {In Logic in Computer Science},

year = {1995},

pages = {444--454}

}

### OpenURL

### Abstract

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 Stone-Pontrjagin 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 category Chu 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 1, 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

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Citation Context ...y 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 [dP... |

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Citation Context ...s, but not abelian groups or nondistributive lattices) treated by Johnstone [Joh82] arise at K = 2. Discussions with M. Barr following an impromptu talk on Chu spaces at MFPS'93 led us to the theorem =-=[Pra93]-=- that every category C of (possibly multisorted) relational structures of total arity k and their homomorphisms was realizable in the category of Chu spaces over 2 k , in the sense that there exists a... |

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Citation Context ...91]. 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 Ne... |

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Citation Context ...ed among these are all categories of relational structures and their homomorphisms standardly defined: posets, groups, graphs, etc., along with topological spaces but not to our knowledge hypergraphs =-=[DW80]. This paper explore-=-s the gamut of Stone duality at K = 2. We show that "continuous function" is equivalent to "homomorphism" for a very general notion of the latter. We recast the proof of our univer... |

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Citation Context ...v93] and Rosebrugh and Wood's appealing one-axiom characterization of Set as the unique choice of V (any category) for which the Yoneda embedding Y V : V ! Set V op has a string of four left adjoints =-=[RW94]-=-. Another rationale for choosing V = Set is its accessibility as a universally understood category, with sets and functions nowadays being taught early on. Whereas category theory is unavoidable for t... |

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Citation Context ...y captured exactly the notion of partial distributive lattice that we had been trying to pin down [Pra92d] as an extension of our earlier notion of event space [Pra92c, Pra92a, Pra92b]. More recently =-=[Pra95]-=- we have arrived at the view of =j and its converse j= as the dual relations of causality and consciousness. The entry a=jx expresses the causal link from physical event a to mental state x whereby th... |

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Citation Context ...rent computation in June of 1992 when we found with our student V. Gupta [Gup93, GP93, Gup94] that they captured exactly the notion of partial distributive lattice that we had been trying to pin down =-=[Pra92d]-=- as an extension of our earlier notion of event space [Pra92c, Pra92a, Pra92b]. More recently [Pra95] we have arrived at the view of =j and its converse j= as the dual relations of causality and consc... |

1 |
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Citation Context ...ormation but need not be localized in time. The dynamic interpretation of Chu spaces has more recently been considerably extended by van Glabbeek and Plotkin under the rubric of configuration systems =-=[VGP95]-=-. The =j relation is the only primitive link needed for this theory of behavior. The causing of events and the extraction of information from states are lesser relations which we have shown [Pra95, p.... |