Talk:Chu space

Untitled

[deleted copy of the original article that I placed here inadvertently] --Vaughan Pratt 22:45, 17 July 2007 (UTC)

Oops

• JA: Sorry, didn't know you were still working on it, will hold off formatting till tomorrow or Monday. Jon Awbrey 05:32, 12 March 2006 (UTC)

On 2

• JA: It might help to bold the "2" when it's used as the name of a domain. Jon Awbrey 17:34, 12 March 2006 (UTC)

It's only used as the name of a set here, namely 2 = {0,1}. Is bold the appropriate way to indicate a set? (I'm fine with bold for categories, e.g. Set.) --Vaughan Pratt 22:50, 17 July 2007 (UTC)

Universality

Out of curiosity, does the proof that every small concrete category is realized in Chu(Set, K) for some set K generalize to showing that every (possibly large) concrete category is realized in Chu(Set, Set)? (I hope what I'm asking makes sense...) -Chinju (talk) 20:21, 20 January 2008 (UTC)

For Chu(X,Y) to be meaningful, Y has to be an object of X. Set is not an object of Set, by Cantor's theorem, but the (large) set of its objects is an object of the category SET of large sets or classes, so Chu(SET,ob(Set)) would make sense. Every category C would then be realizable in Chu(SET,|C|), where |C| denotes the (large) set of morphisms of C, by the same reasoning that every small category C is realizable in Chu(Set,|C|). (Note that every small category C is concrete in the sense that there exists a faithful functor U: C → Set --- take U(c) to be the set of all morphisms to c and U(f) for f: c → d to be the function U(f): U(c) → U(d) taking each x: b → c to the composite fx: b → d; U is faithful because U(f) for f: c → d maps 1_c to f whence distinct morphisms in C(c,d) map 1_c to distinct values. By the same token every category can be considered large concrete in the sense of having a faithful functor to SET. The boundary between small and large can be chosen fairly arbitrarily, and serves mainly to satisfy Cantor's theorem, i.e. avoid Russell's paradox.) --Vaughan Pratt (talk) 16:27, 3 August 2009 (UTC)

drop everything

I complain about the leading section:

“Chu spaces generalize the notion of topological space by dropping the requirements that the set of open sets be closed under union and finite intersection, that the open sets be extensional, and that the membership predicate (of points in open sets) be two-valued.”

It seems this that this definition drops all parts of the definition of a topological space. Then we get nothing, is not it? ;) IMHO it would be better to concentrate on what we append, not what we drop. --Beroal (talk) 13:34, 6 November 2011 (UTC)

(Sorry I didn't see this earlier.) You're absolutely right. At a minimum I could say that the definition of a continuous function remains the same while generalizing other aspects. Much better would be to say what it is explicitly and then point out topological spaces as a very special case, along with vector spaces (though it isn't as obvious that the linear transformations of linear algebra work the same way as continuous functions in point set topology, which AFAIK was first pointed out by Lafont and Streicher in 1991). Let me add that to my list of projects. Vaughan Pratt (talk) 04:55, 17 December 2022 (UTC)

History

Can we add a section on the history of Chu spaces? Where did they come from, and what was the motivation? Reddyuday (talk) —Preceding undated comment added 07:24, 18 July 2012 (UTC)

Certainly worth doing. I'll put that on my bucket list. Vaughan Pratt (talk) 04:56, 17 December 2022 (UTC)

Application: Information flow

I'm tempted to add a section about how Barwise and Seligman use Chu spaces. If r is a modeling relation (true if the state (a model) satisfies the point (a theory)) we obtain a nice framework for comparing formal languages and their respective interpretations. The morphisms in this case are called "infomorphisms", a term that really ought to have at least one Wikipedia search hit. They have some theorems that perhaps I can summarize. Working myself up to writing this paragraph. I may have mangled this but will double check the math before posting anything. Jar354 (talk) 14:59, 24 December 2019 (UTC)

Go for it. IIRC Seligman participated in a workshop on Chu spaces that Valeria de Paiva and I organized. I've never been entirely clear as to whether informorphisms were exactly Chu transforms, partly because Valeria was defining them differently from Chu. Vaughan Pratt (talk) 05:00, 17 December 2022 (UTC)

References