I just wanted to add to what Vaughan said about the provenance of Chu spaces. When I started working on duality in 1975 at the ETH, one prominent example was topologicalvector spaces. So I looked at several books on the subject, one by Grothendieck. They all mentioned a construction of pairs (V,V') of spaces equipped with a bilinear map V x V' --> K (the ground field). This simple example, with its obvious duality led pretty directly to the Chu construction. It was amazing that it turned to not only give and obvious duality but also a natural *-autonomous structure (that Peter Chu studied). At some point I decided to track down the source. It certainly had an air of Grothendieck about it, but no, it turned out to go back to George Mackey's thesis, published in TAMS around 1944, I think. He looked at pairs (V,V') as above and assumed they were extensional but not necessarily separated. He was aware of the duality in the separated extensional case, but didn't pursue it in any detail. So what was the motivation. I can guess that he was thinking that V', as a space of functionals on V, coud replace the topology. Just as in a locally compact abelian group, you could recover the topology if you knew which characters were "admissible". I once wrote him to ask if that was his original motivation, but he died within the ensuing year, not having answered me. But truly, it goes back to Mackey. Michael ----- Original Message ----- From: "Vaughan Pratt" <pratt@cs.stanford.edu> To: "Categories mailing list" <categories@mta.ca> Sent: Sunday, 14 December, 2014 1:12:58 PM Subject: categories: Re: Open problems On 12/12/2014 1:56 PM, Harley Eades III wrote:
Dee Roytenberg?s email pushed me to write an email I have been wanting to write for a bit.
One thing I have been trying to do recently is figure out what the major (and minor) open problems are with respect to applications of category theory to CS. I am very new to this area, and getting an idea of what folks are working on, and what problems people feel are important will help young researchers learns where to concentrate their efforts.
At http://thue.stanford.edu/puzzle.html can be seen a problem about Chu spaces that's been open for close to two decades, and that's been translated during the past three years into Belorussian, Ukrainian, Polish, Slovenian, Czech, French, and Bulgarian by volunteers who apparently found the problem sufficiently interesting to be worth the effort, perhaps motivated by its connection to crossword puzzles. The problem is whether every T1 comonoid in Chu(Set,2) is discrete. This is so for two special cases. 1. Countable comonoids, taking cardinality of a Chu space to be that of its points. 2. Representable comonoids (of any cardinality), in the sense of representation by directed CPOs as per the hierarchy of full embeddings on page 6 of the paper "Comonoids in chu: a large cartesian closed sibling of topological spaces" at http://boole.stanford.edu/pub/comonoids.pdf . T1 DCPOs are of course discrete simply because T1 posets are discrete and the structure of any DCPO is determined by its order via the Scott topology (as distinct from the Alexandroff topology, though they coincide in the T1 case). The problem remains open for uncountable comonoids not representable as DCPOs. ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]