When dealing with the domain theoretic categories (say CPO and $\omega$CPO), I frequently wish for a dictionary for the language of category theory in domain theory. That is, given a concept, say monic arrow, I could look it up in the dictionary and see what are the known characterisations of it in the different domain categories.I realise this wish is too much to hope for, but is there any text or resource approximating it?
What is a good Category Theory-Domain Theory dictionary?
ct.category theory;semantics;denotational semantics;domain theory
The best resource for this is Abramsky and Jung's handbook chapter. I recall they had a table which cross-referenced various constructions and categories of domains, with the entries saying whether the construction worked in that category and what properties it had. However, properties of arrows like being a monic tended not to have terribly slick characterizations, because the availability of flat domains tends to ensure that they are often not terribly different from their set-theoretic counterpart. OTOH, properties that make some use of the order structure (like being an embedding-projection pair) tend to have fairly pretty characterizations.A minor point to watch out for is that there are actually two definitions of CPO in common use! Consumers of domain theory (like me) often prefer to work with omega-chains, since chains are pretty concrete objects; whereas producers of domain theory (like, er, your advisor) tend to prefer to work with directed sets, which are more general and have better algebraic properties. (Offhand I'm not sure if restricting to directed sets having countable base is equivalent to the omega-chain condition.)Something I found very helpful in building this kind of dictionary is to work through the solution of recursive domain equations in some category of things that aren't exactly domains. Two good choices are categories of PERs (eg, in models of polymorphism) and presheaves (eg, for name allocation). Metric spaces are another possibility, but I found them to be too similar to domains to help me build intuition.