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.



approximating categories with continuous functors
As it happens, I just saw a paper about this very subject today -- Martin Hyland's ["Some Reaons for Generalizing Domain Theory"](http://www.dpmms.cam.ac.uk/~martin/Research/Publications/2010/srfg10.pdf), 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"](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.157.6016&rep=rep1&type=pdf). This seems quite accessible to me.


