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# 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 natural and rich structure given by Girard's linear logic [Gir87].

To climb up to this universe of structured objects, we use as our ladder a universe of unstructured or discrete objects, namely ordinary sets and the functions between them. Having done so, we then pull this ladder up after us by representing sets as discrete Chu spaces.

For all practical purposes almost any reasonable understanding of sets and functions will suffice for our development. But it will not hurt to say informally what parts of set theory we make essential use of. Nowhere shall we depend on the existence of infinite sets, though of course if they do exist then one can manufacture infinite Chu spaces from them, essential for representing say the ring of integers or the field of reals.

What we do need is binary cartesian product , along with the function space , the set of all functions from the set A to the set B. We also need the Currying principle, namely the bijection between and which puts each f in the former into correspondence with f' in the latter via f(a,b)=f'(a)(b). And, given two functions , we need to be able 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.]. The case 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 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 objects than previously associated with linear logic.

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Vaughan Pratt
1998-03-14