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I am wondering about approximation and idealization. Specifically, I am wondering if anyone has seen some work on the following. In the semantics of programming languages we find Domains as a place to talk about iteration and approximation. We can define a Scott Topology on the Domain and now our Domain-maps are continuous maps. Next,we can see our Domains as categories and turn the continuous Domain-maps into continuous functors. If we push the idea further, we have continuous functors and a notion of approximation which is now over categories. Lambek ponders the existence of the category of Sets. What about approximations to the category of sets. For that matter, what might it look like to approximate any well-known category like that of manifolds or FDVec? I realize the question is vague. I wish I had a more concrete question, but my grasp of the material is too weak. Naturally, any thoughts on this would be most appreciated.

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Here is the paper which outlines all of my ideas on using continuous functors in physics. cs.mcgill.ca/~bsprot1/EvolvingUniverseFeb24.pdf I realize this is not the place to post this, but some of the posters here might find it of interest. –  Ben Sprott Mar 2 '11 at 17:32
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As it happens, I just saw a paper about this very subject today -- Martin Hyland's "Some Reaons for Generalizing Domain Theory", which is concerned with precisely the generalization you suggest, in order to clarify the semantics of concurrency. This paper was apparently inspired by some work by Cattani and Winskel, but I found their work to be more categorically sophisticated than I could easily digest, and this one to be at about the level I can presently cope with.

EDIT: I also found a paper by Winskel in which he discusses the intuitions underlying his ideas, "Events, Causality, and Symmetry". This seems quite accessible to me.

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Thank you! In fact, these thoughts are coming from my contact with concurrency and distributed systems. To put a point on it, consider a bunch of computers, connected by the internet and thus not sharing a universal clock, and they want to consider themselves as part of a group G. G is a set which contains the names of all the computers. However, agreement is generally untenable in these kinds of asynchronous systems. Thus the concept of a single universe of sets for all these computers is also untenable. Perhaps they can approximate it much like the approximations in agreement protocols. –  Ben Sprott Oct 14 '10 at 15:53
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