Jason (jcreed) wrote,

I checked out the 1979 monograph "*-Autonomous Categories" by Michael Barr from the Pitt math library. I can barely make any headway through the main material, but the appendix by his student Po-Hsiang Chu, which gives the so-called "Chu Construction" of a *-autonomous category from a plain ol' autonomous category (assuming it has pullbacks) and a chosen dualizing object. It turns out this construction yields things that show up all over the place. Mysteriously if you just take Sets with autonomous structure given by just the usual products and exponentials, and choose the dualizing object to be 2k, 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 → 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).
Tags: chu, math
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