Take the 2-minute tour ×   Suppose I want to establish a theory
of the category $C$ (vector spaces or whatever), but what I
really have is $D$, some precisely known category. This is to
say, I know all the axioms of $D$, but I only have an intuition
for $C$ and I want to develop a theory of $C$. I would normally
have diagrams in $C$ by mapping cpos or Domains into $C$. But
instead, I want to do it with $D$. What I do is define the
largest category, $J$, of Domains in $D$. I do this by defining
a dcpo with objects as elements in $D$ and relations as arrows
in $D$. Then any functor will map the domains in $J$ to diagrams
in $C$.  It seems like I am just inserting a category $D$ in the
normal diagram functor $J \rightarrow C$ resulting in $J
\rightarrow D \rightarrow C$ which seems to miss the point of
the exercise. The point of the exercise, I think, is to try to
do a lot of category theory when you have to live in some
category $D$.  We start by saying that we "have access to" all
diagrams in $D$. Further, we