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Mike, if you consider that locally cartesian closed categoriesprovide the person who introduced canonical semantics for dependent type theories thenyou may as well just use sets, for which any function$p:A\to X$ provides a dependent type $A[x]={a|p((a)=x}$.
Not only is this a very dull notion of dependent type,but it gives no account of the name way in which $A{x]$ might depend"display map"continuously" on $x$,something that we probably need to understand in order togive a meaning to the word "recursive".
(Local, relative or ordinary) cartesian closure is needed tointerpret function- or Pi-types, I would like which do not feature in yourquestion. The appropriate arena is a category with some finitelimits and something infinitary to emphasise that capture the whole point recursion.
A class of doing so was to be able display maps is a class of morphisms that isclosed under (composition with isomorphisms and) pullback againstarbitrary maps in the category.
This categorical notion is equivalent to do that of a dependent typetheory in categories that the basic algebraic sense, ie with types, terms, equationsand structural rules.As I believe you are NOTlocally cartesian closed but "relatively" somore comfortable with a categorical language,you can solve your problem in that setting and then use the equivalenceto reformulate it symbolically.
In an LCCCparticular, the class of display maps includes- all isomorphisms iff the type theory includes singleton dependent types;- composites iff the type theory has Sigma types;- inclusions of diagonals and hence all maps are iff the type theory has equality types;- relative cartesian closure corresponds to Pi types.
I had originally interpreted your "recursively dependent" typesto mean an infinite chain of dependencies, and hence of display maps.
see For that you would want the class of displays to be closed undercofiltered limits.
Neel, on the other hand, read it as a fixed point equation,which we can interpret categorically as the fixed point of a functor.
Unsurprisingly, domain theory would be a useful setting in whichto look for models of these situations.my PhD thesisintroduced classes of display maps in order to study dependent typesin domain theory, and you might like to look at the last chapter for investigationsof appropriate notions of displays of domains.
For the theory of display maps and their equivalence with dependenttypes, my thesis was completely superseded by of my book, "Practical Foundations of Mathematics" (CUP 1999)1999).
For the interpretation of dependent types in domain theory,Martin Hyland and Andrew Pitts gave a comprehensive account inThe Theory of Constructions: Categorical Semantics andTopos-Theoretic Models inCategories in Computer Science and Logicedited by John Gray and Andre Scedrov, AMS Contemporary Mathematics 92 (1989).
