**Sets**with autonomous structure given by just the usual products and exponentials, and choose the dualizing object to be 2

^{k}, you get a category that faithfully embeds all of

**Top**for k=2, all of

**Grp**for k=3,

**Rng**and

**Cat**at k=4, and any relational theory at all for some k.

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 → 2

^{k}that extend

**no**tuple from the group multiplication table. Then Chu-continuous functions, allegedly, are just group homomorphisms.

Another interesting paper by Dominic Hughes:

http://www.entcs.org/files/mfps19/83007.pdf

Furthermore, I can't shake the feeling that Chu(

**Sets**, K) is just the comma category

**Sets**↓ (— ⇒ K).