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.
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.