Neat stuff! Apparently Vaughan Pratt has been real big on Chu spaces for a while. I find his writing style moderately difficult — think "Diet Jean-Yves Girard", by which epithet I think I can get away for criticizing him by complimenting him, by the self-same allusion to Girard, for his likely present but slightly obscured genius — but this paper, "The Stone Gamut: A Coordinatization of Mathematics" is still pretty interesting, and gives at least some of the good constructions. If I understand section 6 right, you encode k-relational structures (like models for the theory of groups, which looks like a 3-relational structure, since you have a binary operation and it has an output) as Chu spaces by taking the points to be elements of the carrier A (for a group, the group elements) and the states to be functions A → 2k that extend no tuple from the group multiplication table. Then Chu-continuous functions, allegedly, are just group homomorphisms.
Another interesting paper by Dominic Hughes:
Furthermore, I can't shake the feeling that Chu(Sets, K) is just the comma category
Sets ↓ (— ⇒ K).