## Chu Spaces From the Representational Viewpoint (1998)

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Venue: | Ann. Pure Appl. Logic |

Citations: | 8 - 0 self |

### BibTeX

@ARTICLE{Pratt98chuspaces,

author = {Vaughan Pratt},

title = {Chu Spaces From the Representational Viewpoint},

journal = {Ann. Pure Appl. Logic},

year = {1998},

volume = {96},

pages = {1--3}

}

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### Abstract

We give an elementary introduction to Chu spaces viewed as a set of strings all of the same length. This perspective dualizes the alternative view of Chu spaces as generalized topological spaces, and has the advantage of substituting the intuitions of formal language theory for those of topology. 1 Background Chu spaces provide a simple, uniform, and well-structured approach to the representation of objects that may possess algebraic, relational, or other structure, and that can transform into one another in ways that respect that structure. Chu spaces are simple by virtue of being merely a rectangular array, with no further machinery. They are uniform in the sense that all transformable objects, whether sets, groups, Boolean algebras, vector spaces, or manifolds, are representable by Chu spaces within the same framework, and hence can coexist in a single typeless universe of mathematical objects. And they are well-structured in that this seemingly featureless universe in fact has a na...

### Citations

377 | Basic concepts of enriched category theory, volume 64
- Kelly
- 1982
(Show Context)
Citation Context ...dea of representing duality as a contravariant pair of morphisms goes back to G. Mackey [Mac45]. 2 This idea was abstracted by M. Barr to Chu spaces enriched (in the sense of enriched category theory =-=[Kel82]-=-) in a symmetric monoidal category V and studied by his student Peter Chu [Bar79, App.]. The case V = Set of ordinary Chu spaces, the kind we treat here, was first studied in detail by Lafont and Stre... |

87 | Autonomous categories and linear logic - Barr - 1991 |

54 | Origins of the Calculus of Binary Relations
- Pratt
- 1992
(Show Context)
Citation Context ...ing r to A times that intersection. One can then show that a function from (A; R) to (B; S) is a homomorphism if and only if it is a continuous function between the respective representing Chu spaces =-=[Pra93]-=-. The construction generalizes to multiple relations by combining them into one super-relation with arity the sum of its constituent arities. It further generalizes to multiple sorts by combining them... |

48 |
Games Semantics for Linear Logic
- Lafont, Streicher
- 1991
(Show Context)
Citation Context ... symmetric monoidal category V and studied by his student Peter Chu [Bar79, App.]. The case V = Set of ordinary Chu spaces, the kind we treat here, was first studied in detail by Lafont and Streicher =-=[LS91] under the-=- rubric of games, and by Brown, Gurr, and de Paiva [BGdP91], and Blass [Bla95], who have treated a "lax-continuous" variant in which the adjointness condition defining continuity is relaxed ... |

45 |
Autonomous Categories, volume 752
- Barr
- 1979
(Show Context)
Citation Context ...representation of x. It is then straightforward to show that this representation is faithful, full, and concrete. 7 Linear Logic Chu spaces and their continuous functions form a *-autonomous category =-=[Bar79]-=-, one that is self-dual and symmetric monoidal closed. This places them squarely in the domain of discourse of Girard's linear logic [Bar91,Gir87]. In this section we define the multiplicative fragmen... |

32 | Gages accept concurrent behavior
- Gupta, Pratt
- 1993
(Show Context)
Citation Context ...ntness condition defining continuity is relaxed from an equality to an inequality. Our own interest in Chu spaces originated in their application to the representation of generalized event structures =-=[GP93]-=-, but we have since found them also of interest as universal objects [Pra95,Pra96], broadening the denotational semantics of linear logic to a much larger, in fact universal, class of mathematical obj... |

30 | The Stone gamut: A coordinatization of mathematics
- Pratt
- 1995
(Show Context)
Citation Context ..., with \Sigma still 2, the maximal structure with which we may equip \Sigma X is that of a complete atomic Boolean algebra. If we ignore questions of concreteness, this remains true for larger \Sigma =-=[Pra95]-=-. 5 Abstract Structure In this section we shall identify the states of any extensional Chu space (A; r; X) with the dual words representing them. Thus instead of an extensional Chu space being (A; r; ... |

28 | A linear specification language for Petri nets
- Brown, Gurr, et al.
- 1991
(Show Context)
Citation Context ... [Bar79, App.]. The case V = Set of ordinary Chu spaces, the kind we treat here, was first studied in detail by Lafont and Streicher [LS91] under the rubric of games, and by Brown, Gurr, and de Paiva =-=[BGdP91], and Blas-=-s [Bla95], who have treated a "lax-continuous" variant in which the adjointness condition defining continuity is relaxed from an equality to an inequality. Our own interest in Chu spaces ori... |

15 | How comprehensive is the category of semigroup - HedrlĂn, Lambek - 1969 |

9 |
Infinite dimensional vector spaces
- Mackey
- 1945
(Show Context)
Citation Context ...ble to form the subset of A on which f and g agree. The history of Chu spaces is roughly as follows. The basic idea of representing duality as a contravariant pair of morphisms goes back to G. Mackey =-=[Mac45]-=-. 2 This idea was abstracted by M. Barr to Chu spaces enriched (in the sense of enriched category theory [Kel82]) in a symmetric monoidal category V and studied by his student Peter Chu [Bar79, App.].... |

6 | Towards full completeness of the linear logic of Chu spaces. Mathematical foundations of programming semantics
- Pratt
- 1997
(Show Context)
Citation Context ... . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ B, as A ? \GammaffiB. In =-=[Pra97]-=- we make a start on the project of proving full completeness of linear logic interpreted over Chu spaces, by showing that for formulas of multiplicative linear logic having at most two occurrences of ... |

5 | Universal categories - TrnkovĂˇ - 1966 |

3 | Broadening the denotational semantics of linear logic
- Pratt
- 1996
(Show Context)
Citation Context ... of F (b 0 ) to states of F (b). That this is indeed the adjoint follows immediately from (f ; g); h = f ; (g; h), associativity of composition. That F is full and faithful is a nice exercise (or see =-=[Pra96]-=-). That F is concrete (commutes with the forgetful functor to Set) holds when we take as the forgetful functor for small categories the functor that sends each object to the set of all morphisms to th... |

1 |
A category arising in linear logic, complexity theory, and set theory
- Blass
- 1995
(Show Context)
Citation Context ...case V = Set of ordinary Chu spaces, the kind we treat here, was first studied in detail by Lafont and Streicher [LS91] under the rubric of games, and by Brown, Gurr, and de Paiva [BGdP91], and Blass =-=[Bla95], who have-=- treated a "lax-continuous" variant in which the adjointness condition defining continuity is relaxed from an equality to an inequality. Our own interest in Chu spaces originated in their ap... |