Mike, if you consider that locally cartesian closed categories
provide the canonical semantics for dependent type theories then
you 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 way in which $A{x]$ might depend
"continuously" on $x$,
something that we probably need to understand in order to
give a meaning to the word "recursive".

(Local, relative or ordinary) cartesian closure is needed to
interpret function- or Pi-types, which do not feature in your
question. The appropriate arena is a category with some finite
limits and something infinitary to capture the recursion.

A **class of display maps** is a class of morphisms that is
closed under (composition with isomorphisms and) pullback against
arbitrary maps in the category.

This categorical notion is equivalent to that of a dependent type
theory in the basic algebraic sense, ie with types, terms, equations
and structural rules.
As I believe you are more comfortable with a categorical language,
you can solve your problem in that setting and then use the equivalence
to reformulate it symbolically.

In particular, 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 iff the type theory has
equality types;
- relative cartesian closure corresponds to Pi types.

I had originally interpreted your "recursively dependent" types
to mean an infinite chain of dependencies, and hence of display maps.
For that you would want the class of displays to be closed under
cofiltered 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 which
to look for models of these situations.
Indeed
my PhD thesis
introduced classes of display maps in order to study dependent types
in domain theory,
and you might like to look at the last chapter for investigations
of appropriate notions of displays of domains.

For the theory of display maps and their equivalence with dependent
types, my thesis was completely superseded by
Chapter VIII
of my book, "Practical Foundations of Mathematics" (CUP 1999).

For the interpretation of dependent types in domain theory,
Martin Hyland and Andrew Pitts gave a comprehensive account in
their paper
*The Theory of Constructions: Categorical Semantics and
Topos-Theoretic Models* in
*Categories in Computer Science and Logic*
edited by John Gray and Andre Scedrov,
AMS *Contemporary Mathematics* **92** (1989).