Universal topology: Representing concrete objects by Chu spaces

Vaughan Pratt – 29 July 1998

A Chu space (A,r,X) over an alphabet K consists of a carrier A consisting of points a,a',..., a "cocarrier" X consisting of dual points or states x,x',..., and an "interaction matrix" r:AxX->K. Morphisms are defined by analogy with continuous functions between topological spaces, with X as the set of open sets and r(a,x) as the "degree of membership" of point a in "open set" x; equivalently they may be understood as abstract adjunctions, the "right adjoint" to f:A->B being the map g:Y->X sending open sets of B to open sets of A. As such Chu spaces amount to generalized topological spaces, with no closure conditions on the open sets and any (fixed) number of degrees of membership. For ordinary Chu spaces, A, X, and K are sets and r is a function, while the V-enriched generalization takes these from any symmetric monoidal closed V. The category of V-enriched spaces with alphabet (dualizing object) k, introduced by Barr and his student Chu, is denoted Chu(V,k). We show that ordinary Chu spaces have a universal character indicated by several concrete full embeddings. The category of n-ary relational structures and their homomorphisms so embeds in Chu(Set,2^n). Any small category C embeds fully in Chu(Set,|C|) (K = the set of arrows of C). And any small concrete category embeds fully and concretely in Chu(Set,K) where K is the set of all elements of C, i.e. the disjoint union of the underlying sets of the objects of C.