FOM: The notion of Chu space

[Vaughan R. Pratt]

This is the sketch of Chu spaces promised in my previous message.

Chu spaces generalize the notion of topological space in two ways: the
usual closure conditions on open sets are dropped, and more than two
degrees of membership in the open sets are permitted ("fuzzy open
sets").  The definition of continuity can remain unchanged provided it
is worded in the obvious way so as to accommodate fuzzy membership.

This generalization of topological space can mimic (i.e. fully,
faithfully, and concretely represent) any relational structure, with or
without relational structure.  Even more surprisingly, they can also
mimic the objects of *any* small category.  (Being essentially just a
(constructively) reflexive transitive multigraph, a category can have
pretty arbitrary structure.)

The relevance here of linear logic is that it is the natural logic of
Chu spaces.  The essence of multiplicative linear logic is already
found in *-autonomous categories, introduced by Mike Barr in 1979,
several years before linear logic was described by Girard.  A major
part of Girard's contribution has been to develop its attractive proof
theory.

Vaughan Pratt