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?

Naturally, any thoughts on this would be most appreciated.

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.